# Generalization of Bair spaces using diagonal

Evgenii Reznichenko<EMAIL_ADDRESS>Department of General Topology and
Geometry, Mechanics and Mathematics Faculty, M. V. Lomonosov Moscow State
University, Leninskie Gory 1, Moscow, 199991 Russia

###### Abstract

Generalizations of Baire spaces are constructed with the help of diagonal of
square.

###### keywords:

Baire space , nonmeagre space , topological games , generalization of Baire
space , diagonal of square ,

###### MSC:

[2010] 54B10 , 54C30 , 54C05 , 54C20

##  1 Introduction

A space $X$ is called Baire (nonmeager) if for any family $(U_{n})_{n}$ of
open dense subsets of $X$ the intersection $G=\bigcap_{n}U_{n}$ is dense in
$X$ (not empty).

Baire spaces play an important role in mathematics. The strongest and most
interesting results are obtained when additional constraints on the space are
imposed, such as metrizability. In [1], enhancements of the Baire property are
defined using games of the Banach–Mazur type.

In [2] $\Delta$-Baire property was found, which implies the continuity of
operations in groups. The $\Delta$-Baire properties are defined in terms of
diagonal semi-neighborhoods (see section 7). A game was found, with the help
of which $\Delta$-Baire spaces were constructed.

This article considers new classes of spaces defined by diagonal semi-
neighborhoods (see Section 9) and links these properties to the
generalizations of Baire obtained by games (see Section 11).

The results of this article and of [1] were used in [3] to study the
continuity of group operations in right-topological groups.

##  2 Definitions and notation

The sign $\coloneqq$ will be used for equality by definition.

The family of all subsets of a set $X$ is denoted by $\operatorname{Exp}(X)$.
The family of all nonempty subsets of a set $X$ is denoted by
$\operatorname{{Exp}}_{*}(X)$:
$\operatorname{{Exp}}_{*}(X)\coloneqq\operatorname{Exp}(X)\setminus\\{\varnothing\\}$.

If $B$ is a subset of a set $A$, then we denote by $B^{c}=A\setminus B$ the
complement to $A$. We use this notation in situations where it is clear from
the context which set $A$ is meant.

An indexed set $x=(x_{\alpha})_{\alpha\in A}$ is a function on $A$ such that
$x(\alpha)=x_{\alpha}$ for $\alpha\in A$. If the elements of an indexed set
${\mathcal{X}}=(X_{\alpha})_{\alpha\in A}$ are themselves sets, then
${\mathcal{X}}$ is also called an indexed family of sets; such an
${\mathcal{X}}$ is a function on $A$ such that
${\mathcal{X}}(\alpha)=X_{\alpha}$ for ${\alpha\in P}$.

We denote by $\operatorname{Aut}(X)$ the set of all homeomorphisms of the
space $X$ onto itself.

A subset $M$ of a topological space $X$ is called locally dense (nearly open
or preopen) if $M\subset\operatorname{Int}\overline{M}$.

Let $M\subset X$. If $M$ is the union of a countable number of nowhere dense
sets, then $M$ is called a meager set. Nonmeager sets are called of the second
Baire category sets. A subset of $M$ is called residual (comeager) if
$X\setminus M$ is a meager set.

A family $\nu$ of nonempty subsets of $X$ is called a $\pi$-net if for any
open nonempty $U\subset X$ there exists an $M\in\nu$ such that $M\subset U$.

A $\pi$-network consisting of open sets is called a $\pi$-base.

A subset $U\subset X$ is called regular open if
$U=\operatorname{Int}{\overline{U}}$.

A space $X$ is called quasi-regular if for every nonempty open $U\subset X$
there exists a nonempty open $V\subset X$such that $\overline{V}\subset U$.

A space $X$ is called semiregular if $X$ has a base consisting of regular open
sets.

A space $X$ is called $\pi$-semiregular [4] (or nearly regular [5]) if $X$ is
a $\pi$-base consisting of regular open sets.

For a cardinal $\tau$, a set $G\subset X$ is called a set of type $G_{\tau}$
if $G$ is the intersection of $\tau$ open sets. A space $X$ is called absolute
$G_{\tau}$ if $X$ is of type $G_{\tau}$ in some compact extension.

A space $X$ is regular at a point $x\in X$ if for any neighborhood $U$ of the
point $x$ there exists a neighborhood $V\ni x$ such that $\overline{V}\subset
U$.

A space $X$ is semiregular at a point $x\in X$ if there is a base at the point
$x$ consisting of regular open sets.

A space $X$ is feebly compact if any locally finite family of open sets is
finite.

For $\gamma\subset\operatorname{Exp}(X)$ and $x\in X$ we denote

$\displaystyle\operatorname{St}(x,\gamma)$
$\displaystyle\coloneqq\\{U\in\gamma\,:\,x\in\gamma\\},$
$\displaystyle\operatorname{st}(x,\gamma)$
$\displaystyle\coloneqq\bigcup\operatorname{St}(x,\gamma).$

A space $X$ is called developable if there exists a sequence of open covers
$(\gamma_{n})_{n\in\omega}$ such that for any $x\in X$ the family
$\operatorname{st}(x,\gamma_{n})$ is a base at the point $x$.

A family ${\mathcal{B}}$ of open nonempty sets in $X$ is called an outer base
of $M\subset X$ if $M\subset U$ for each $U\in{\mathcal{B}}$ and for each open
$W\supset M$ there exists a $U\in{\mathcal{B}}$ such that $M\subset U\subset
W$.

If $(M_{n})_{n\in\omega}$ is a sequence of subsets of a space $X$, then the
set

$\displaystyle\operatorname{\overline{lt}}_{n\in\omega}M_{n}\coloneqq\\{x\in
X:\ $ $\displaystyle|\\{n\in\omega\,:\,U\cap M_{n}\neq\varnothing\\}|=\omega$
$\displaystyle\text{ for any neighborhood }U\text{ of the point }x\\}$

is called the upper limit of the sequence of sets $(M_{n})_{n\in\omega}$.

If $(x_{n})_{n\in\omega}$ is a sequence of points in a space $X$, then we
denote

$\displaystyle\operatorname{\overline{lt}}_{n\in\omega}x_{n}$
$\displaystyle\coloneqq\operatorname{\overline{lt}}_{n\in\omega}\\{x_{n}\\}.$

We denote by $\operatorname{\beta}\omega$ the space of ultrafilters on
$\omega$, the Stone-Čech extension of the discrete space $\omega$. We denote
by $\omega^{*}=\operatorname{\beta}\omega\setminus\omega$ the set of
nonprincipal ultrafilters.

Let $(x_{n})_{n\in\omega}$ be a sequence of points in a space $X$ and let
$p\in\omega^{*}$ be a nonprincipal ultrafilter. A point $x\in X$ is called the
$p$-limit of a sequence $(x_{n})_{n\in\omega}$ if $\\{n\in\omega\,:\,x_{n}\in
U\\}\in p$ for any neighborhood $U$ of $x$. We will write
$x={\textstyle\lim_{p}}x_{n}={\textstyle\lim_{p}}(x_{n})_{n\in\omega}$ for the
$p$-limit $x$.

##  3 Modifications of the Banach-Mazur game

In this section we use topological games. Basic concepts can be found in [6,
7, 8, 2]. The terminology and notation correspond to [1].

Let $G$ be a game in which player $\alpha$ has a winning strategy. Let us call
this game $G$ $\alpha$-favorable. If there is no such strategy, then the $G$
$\alpha$-unfavorable game.

If the definition of a game $G$ depends on only one parameter, namely, some
space $X$, that is, the game $G=\Gamma(X)$, then we write space $X$
$(\alpha,\Gamma)$-favorable if the game $\Gamma(X)$ is $\alpha$-favorable and
space $X$ is $(\alpha,\Gamma)$-unfavorable if the game is $\Gamma(X)$ $\alpha$
is unfavorable.

Let $(X,{\mathcal{T}})$ be a space,
${{\mathcal{T}}^{*}}={\mathcal{T}}\setminus\\{\varnothing\\}$,
$U\in{{\mathcal{T}}^{*}}$. We denote

$\displaystyle{\mathfrak{V}}(X)$
$\displaystyle\coloneqq\\{(V_{n})_{n\in\omega}\in{{\mathcal{T}}^{*}}^{\omega}\,:\,V_{n+1}\subset
V_{n}\text{ for }{n\in\omega}\\},$ $\displaystyle\Upsilon(U)$
$\displaystyle\coloneqq\\{V\in{{\mathcal{T}}^{*}}\,:\,V\subset U\\}.$

### The games $BM(X,{\mathcal{V}})$ and $MB(X,{\mathcal{V}})$ [1]

Let ${\mathcal{V}}\subset{\mathfrak{V}}(X)$. There are two players, $\alpha$
and $\beta$. These games differ the first move of player $\alpha$. On the
first move, player $\alpha$ chooses $U_{0}=X$ in the game
$BM(X,{\mathcal{V}})$ and $U_{0}\in\Upsilon(X)$ in the game
$MB(X,{\mathcal{V}})$. Player $\beta$ chooses $V_{0}\in\Upsilon(U_{0})$. On
the $n$th move, $\alpha$ chooses $U_{n}\in\Upsilon(V_{n-1})$ and $\beta$
chooses $V_{n}\in\Upsilon(U_{n})$. After a countable number of moves, the
winner is determined: player $\alpha$ wins if
$(V_{n})_{n\in\omega}\in{\mathcal{V}}$.

### The games $OD(X,{\mathcal{N}},\mathcal{W})$ and
$DO(X,{\mathcal{N}},\mathcal{W}$ [1]

Let ${\mathcal{N}}$ be a $\pi$-net of $X$ and let
$\mathcal{W}\subset{\mathfrak{W}}(X)$. There are two players, $\alpha$ and
$\beta$. These games differ in the first move of player $\alpha$. On the first
move, player $\alpha$ chooses $U_{0}=X$ in the game
$OD(X,{\mathcal{N}},\mathcal{W})$ and $U_{0}\in\Upsilon(X)$ in the game
$DO(X,{\mathcal{N}},\mathcal{W})$. Player $\beta$ chooses
$V_{0}\in\Upsilon(U_{0})$ and $M_{0}\in{\mathcal{N}}$, $M_{0}\subset U_{0}$.
On the $n$th move $\alpha$ chooses $U_{n}\in\Upsilon(V_{n-1})$ and $\beta$
chooses $V_{n}\in\Upsilon(U_{n})$ and $M_{n}\in{\mathcal{N}}$, $M_{n}\subset
U_{n}$. After a countable number of moves, the winner is determined: player
$\alpha$ wins if $(V_{n},M_{n})_{n\in\omega}\in\mathcal{W}$.

Games of the form $BM$ ($MB$) are a special case of $OD$ ($DO$) games with an
appropriate choice of $\mathcal{W}$ [1].

##  4 Generalization of the Baire properties and nonmeager through games

Let $(X,{\mathcal{T}})$ be a space,
${{\mathcal{T}}^{*}}={\mathcal{T}}\setminus\\{\varnothing\\}$. We denote

$\displaystyle{\mathfrak{V}}_{BM}(X)$
$\displaystyle\coloneqq\\{(V_{n})_{n\in\omega}\in{\mathfrak{V}}(X)\,:\,\bigcap_{{n\in\omega}}V_{n}\neq\varnothing\\},$
$\displaystyle{\mathfrak{V}}_{BM}^{*}(X)$
$\displaystyle\coloneqq{\mathfrak{V}}(X)\setminus{\mathfrak{V}}_{BM}(X),$
$\displaystyle{\mathfrak{V}}_{R}(X)$
$\displaystyle\coloneqq\\{(V_{n})_{n\in\omega}\in{\mathfrak{V}}(X)\,:\,\overline{V_{n+1}}\subset
V_{n}\text{ for }{n\in\omega}\\}.$

The game $BM(X)=BM(X,{\mathfrak{V}}_{BM}(X))$ is the classical Banach–Mazur
game.

######  Theorem 1 (Banach–Oxtoby [6]; see also [7, 8]).

Let $X$ be a space.

  * (1)

$X$ is Baire if and only if $BM(X)$ is $\beta$-unfavorable;

  * (2)

$X$ is nonmeager if and only if $MB(X)$ is $\beta$-unfavorable.

######  Definition 1 ([1]).

Let $X$ be a space, ${\mathcal{V}}\subset{\mathfrak{V}}(X)$ and
${\mathcal{V}}^{*}={\mathcal{V}}\cup{\mathfrak{V}}_{BM}^{*}(X)$. We call the
space $X$

  * 1.

$\Gamma^{\scriptscriptstyle{BM}}({\mathcal{V}})$-nonmeager if
$MB(X,{\mathcal{V}})$ is $\beta$-unfavorable;

  * 2.

$\Gamma^{\scriptscriptstyle{BM}}({\mathcal{V}})$-Baire if
$BM(X,{\mathcal{V}})$ is $\beta$-unfavorable;

  * 3.

$\Gamma^{\scriptscriptstyle{BM}}({\mathcal{V}})$-space if
$BM(X,{\mathcal{V}}^{*})$ is $\alpha$-favourable.

We set

$\displaystyle{\mathfrak{W}}_{e}(X)\coloneqq\\{(V_{n},M_{n})_{n\in\omega}\in{\mathfrak{W}}(X):$
$\displaystyle\ \bigcap_{{n\in\omega}}V_{n}=\varnothing\\}.$

######  Definition 2 ([1]).

Let $X$ be a space, ${\mathcal{N}}$ be a $\pi$-net $X$,
$\mathcal{W}\subset{\mathfrak{W}}(X)$ and
$\mathcal{W}^{*}=\mathcal{W}\cup{\mathfrak{W}}_{e}(X)$. We call the space $X$

  * 1.

$\Gamma^{\scriptscriptstyle{OD}}({\mathcal{N}},\mathcal{W})$-nonmeager if
$DO(X,{\mathcal{N}},\mathcal{W})$ is $\beta$-unfavorable;

  * 2.

$\Gamma^{\scriptscriptstyle{OD}}({\mathcal{N}},\mathcal{W})$-Baire if
$OD(X,{\mathcal{N}},\mathcal{W})$ is $\beta$-unfavorable;

  * 3.

$\Gamma^{\scriptscriptstyle{OD}}({\mathcal{N}},\mathcal{W})$-space if
$OD(X,{\mathcal{N}},\mathcal{W}^{*})$ is $\alpha$-favourable.

For $\mathcal{W}\subset{\mathfrak{W}}(X)$ we set

$\displaystyle{\mathfrak{V}}_{w}(X,{\mathcal{N}},\mathcal{W})\coloneqq\\{(V_{n})_{n\in\omega}$
$\displaystyle\in{\mathfrak{V}}(X):\text{ if }M_{n}\subset V_{n}\text{ and
}M_{n}\in{\mathcal{N}}$ $\displaystyle\text{ for }{n\in\omega}\text{, then
}(V_{n},M_{n})_{n\in\omega}\in\mathcal{W}\\}.$

We set

$\displaystyle{\mathfrak{W}}_{l}(X)\coloneqq\\{(V_{n},M_{n})_{n\in\omega}\in{\mathfrak{W}}(X):\operatorname{\overline{lt}}_{{n\in\omega}}M_{n}\cap\bigcap_{{n\in\omega}}V_{n}\neq\varnothing\\}.$

We define the family ${\mathfrak{W}}_{k}(X)\subset{\mathfrak{W}}(X)$ as
follows: a sequence $(V_{n},M_{n})_{n\in\omega}\in{\mathfrak{W}}(X)$ belongs
to ${\mathfrak{W}}_{k}(X)$ if the condition is met:

  * there is a sequence $(x_{n})_{n\in\omega}$ satisfying the condition:

    * $(LSQ)$

$x_{n}\in M_{n}$ for ${n\in\omega}$ and for each $p\in\omega^{*}$ there is a
$x\in\bigcap_{{n\in\omega}}V_{n}$ such that $x={\textstyle\lim_{p}}x_{n}$.

Note that if $X$ is a regular space, then the condition $(LSQ)$ is equivalent
to the condition

  *     * $(LSQ)^{\prime}$

$x_{n}\in M_{n}$ for ${n\in\omega}$, the subspace
$\overline{\\{x_{n}\,:\,{n\in\omega}\\}}$ is compact and
$\operatorname{\overline{lt}}_{{n\in\omega}}x_{n}\subset\bigcap_{{n\in\omega}}V_{n}$.

We denote

$\displaystyle{\mathrm{L}}((V_{n},M_{n})_{n\in\omega})=\\{(x_{n})_{n\in\omega}\in
X^{\omega}:\ $ $\displaystyle(x_{n})_{n\in\omega}\text{ satisfies the}$
$\displaystyle\text{ condition }(LSQ)\\}$

and say that $(V_{n},M_{n})_{n\in\omega}\in{\mathfrak{W}}_{k}(X)$ if and only
if ${\mathrm{L}}((V_{n},M_{n})_{n\in\omega})\neq\varnothing$.

We set

$\displaystyle{\mathcal{N}}_{o}(X)$
$\displaystyle\coloneqq{{\mathcal{T}}^{*}},$
$\displaystyle{\mathcal{N}}_{p}(X)$ $\displaystyle\coloneqq\\{\\{x\\}\,:\,x\in
X\\},$ $\displaystyle{\mathfrak{V}}_{o}(X)$
$\displaystyle\coloneqq{\mathfrak{V}}_{w}(X,{\mathcal{N}}_{o}(X),{\mathfrak{W}}_{l}(X)),$
$\displaystyle{\mathfrak{V}}_{p}(X)$
$\displaystyle\coloneqq{\mathfrak{V}}_{w}(X,{\mathcal{N}}_{p}(X),{\mathfrak{W}}_{l}(X)),$
$\displaystyle{\mathfrak{V}}_{f}(X)\coloneqq\\{(U_{n})_{n\in\omega}\in{\mathfrak{V}}_{BM}(X):\
$ $\displaystyle\text{for some }x\in\bigcap_{{n\in\omega}}U_{n}$
$\displaystyle\text{ the family }(U_{n})_{n\in\omega}\text{ forms a base at
the point }x\\},$
$\displaystyle{\mathfrak{V}}_{k}(X)\coloneqq\\{(U_{n})_{n\in\omega}\in{\mathfrak{V}}_{BM}(X):\
$ $\displaystyle\bigcap_{{n\in\omega}}U_{n}=M\text{ is compact and the family
}(U_{n})_{n\in\omega}$ $\displaystyle\text{is an outer base of the set }M\\}.$

Note that

  * 1.

$(V_{n})_{n\in\omega}\in{\mathfrak{V}}_{o}(X)$ if and only if
$(V_{n})_{n\in\omega}\in{\mathfrak{V}}_{BN}(X)$ and for any sequence of open
nonempty sets $(M_{n})_{n\in\omega}$ such that $M_{n}\subset V_{n}$ for
${n\in\omega}$, we have
$\operatorname{\overline{lt}}_{{n\in\omega}}M_{n}\subset\bigcap_{n\in\omega}V_{n}$;

  * 2.

$(V_{n})_{n\in\omega}\in{\mathfrak{V}}_{p}(X)$ if and only if
$(V_{n})_{n\in\omega}\in{\mathfrak{V}}_{BN}(X)$ and for any sequence of points
$(x_{n})_{n\in\omega}$ such that $x_{n}\in V_{n}$ for ${n\in\omega}$, we have
$\operatorname{\overline{lt}}_{{n\in\omega}}x_{n}\subset\bigcap_{n\in\omega}V_{n}$.

For $r\in\\{o,p,f,k\\}$ we define the games

$\displaystyle BM_{r}(X)$ $\displaystyle\coloneqq BM(X,{\mathcal{V}}),$
$\displaystyle MB_{r}(X)$ $\displaystyle\coloneqq MB(X,{\mathcal{V}}),$
$\displaystyle BM_{r}^{*}(X)$ $\displaystyle\coloneqq
BM(X,{\mathcal{V}}^{*}),$ $\displaystyle MB_{r}^{*}(X)$
$\displaystyle\coloneqq MB(X,{\mathcal{V}}^{*}),$

where

$\displaystyle{\mathcal{V}}$ $\displaystyle={\mathfrak{V}}_{r}(X),$
$\displaystyle{\mathcal{V}}^{*}$
$\displaystyle={\mathcal{V}}\cup{\mathfrak{V}}_{BM}^{*}(X).$

For $t\in\\{o,p\\}$ and $q\in\\{l,k\\}$ we define the games

$\displaystyle OD_{t,q}(X)$ $\displaystyle\coloneqq
OD(X,\mathcal{N},\mathcal{W}),$ $\displaystyle DO_{t,q}(X)$
$\displaystyle\coloneqq DO(X,\mathcal{N},\mathcal{W}),$ $\displaystyle
OD_{t,q}^{*}(X)$ $\displaystyle\coloneqq OD(X,\mathcal{N},\mathcal{W}^{*}),$
$\displaystyle DO_{t,q}^{*}(X)$ $\displaystyle\coloneqq
DO(X,\mathcal{N},\mathcal{W}^{*}),$

where

$\displaystyle\mathcal{N}$ $\displaystyle={\mathcal{N}}_{t}(X),$
$\displaystyle\mathcal{W}$ $\displaystyle={\mathfrak{W}}_{q}(X),$
$\displaystyle\mathcal{W}^{*}$
$\displaystyle=\mathcal{W}\cup{\mathfrak{W}}_{e}(X).$

######  Definition 3 (Definition 6 [1]).

Let $X$ be a space, $r\in\\{o,p,f,k\\}$. We call the space $X$

  * 1.

$\Gamma^{\scriptscriptstyle{BM}}_{r}$-nonmeager if $X$ is
$(\beta,MB_{r})$-unfavourable;

  * 2.

$\Gamma^{\scriptscriptstyle{BM}}_{r}$-Baire if $X$ is
$(\beta,BM_{r})$-unfavourable;

  * 3.

$\Gamma^{\scriptscriptstyle{BM}}_{r}$-space if $X$ is
$(\alpha,BM_{r})$-favourable.

The class of $\Gamma^{\scriptscriptstyle{BM}}_{r}$-spaces will be denoted as
$\Gamma^{\scriptscriptstyle{BM}}_{r}$.

Let $t\in\\{o,p\\}$ and $q\in\\{l,k\\}$. We call the space $X$

  * 1.

$\Gamma^{\scriptscriptstyle{OD}}_{t,q}$-nonmeager if $X$ is
$(\beta,DO_{t,q})$-unfavourable;

  * 2.

$\Gamma^{\scriptscriptstyle{OD}}_{t,q}$-Baire if $X$ is
$(\beta,OD_{t,q})$-unfavourable;

  * 3.

$\Gamma^{\scriptscriptstyle{OD}}_{t,q}$-space if $X$ is
$(\alpha,OD_{t,q})$-favourable.

The class of $\Gamma^{\scriptscriptstyle{OD}}_{t,q}$-spaces will be denoted as
$\Gamma^{\scriptscriptstyle{OD}}_{t,q}$.

######  Definition 4 ([1]).

Let $X$ be a space, $x\in X$. Let $t\in\\{o,p,k,f\\}$. We say that a point $x$
is a $q_{t}$-point if there exists
$(V_{n})_{n\in\omega}\in{\mathfrak{V}}_{t}(X)$ such that
$x\in\bigcap_{n\in\omega}V_{n}$.

Recall that a point $x\in X$ is called a $q$-point if there exists
$(V_{n})_{n\in\omega}\in{\mathfrak{V}}_{BM}(X)$ such that
$x\in\bigcap_{n\in\omega}V_{n}$ and any sequence $(x_{n})_{n\in\omega}$ such
that $x_{n}\in V_{n}$ for ${n\in\omega}$, accumulates to some point; see [9].
If $X$ is a regular space, then $x$ is a $q$-point if and only if $x$ is a
$q_{p}$-point. Spaces of point-countable type are precisely spaces in which
each point is a $q_{k}$-point. A point is a $q_{f}$-point if and only if this
point has a countable base.

The definitions imply the following assertion.

######  Proposition 1 (Proposition 12 [1]).

Let $X$ be a space and let $t\in\\{o,p,k,f\\}$. If $X$ is a
$\Gamma^{\scriptscriptstyle{BM}}_{t}$-nonmeager
($\Gamma^{\scriptscriptstyle{BM}}_{t}$-Baire) space, then there are
$q_{t}$-points in $X$ (the set of $q_{t}$-points is dense in $X$).

######  Definition 5 ([2]).

Let $X$ be a space and let $Y\subset X$. We call $Y$ $C$-dense if
$\overline{Y}=X$ and for any countable family $\gamma$ of open subsets of $X$
the family $\gamma$ is locally finite if and only if the family $\\{U\cap
Y\,:\,U\in\gamma\\}$ is locally finite in $Y$.

For a Tychonoff $X$, $Y$ is $C$-dense in $X$ if and only if $Y$ is dense in
$X$ and is $C$-embedded in $X$.

######  Proposition 2 (Proposition 13 [1]).

If $X$ is a quasi-regular space, then $Y\subset X\subset\overline{Y}$. Let
$\Gamma\in\\{\Gamma^{\scriptscriptstyle{BM}}_{o},\Gamma^{\scriptscriptstyle{OD}}_{o,l}\\}$.

  * (1)

If $Y$ is a $\Gamma$-nonmeager ($\Gamma$-Baire) space, then $X$ is a
$\Gamma$-nonmeager ($\Gamma$-Baire) space.

  * (2)

Let $X$ be a quasi-regular space and $Y$ be $C$-dense in $X$. A space $Y$ is
$\Gamma$-nonmeager ($\Gamma$-Baire) if and only if $Y$ is $\Gamma$-nonmeager
($\Gamma$-Baire).

######  Proposition 3 (Proposition 14 [1]).

Let $X$ be a space. In the diagrams below, the arrow

$A\to B$

means that

  * (1)

If $X$ is an $A$-nonmeager space, then $X$ is a $B$-nonmeager space;

  * (2)

if $X$ is an $A$-Baire space, then $X$ is a $B$-Baire space;

  * (3)

if $X$ is an $A$-space, then $X$ is a $B$-space.

The bottom arrow means that $A$-nonmeager and $A$-Baire imply nonmeager and
Baire.

${\Gamma^{\scriptscriptstyle{BM}}_{f}}$${\Gamma^{\scriptscriptstyle{BM}}_{k}}$${\Gamma^{\scriptscriptstyle{BM}}_{o}}$${\Gamma^{\scriptscriptstyle{BM}}_{p}}$${\Gamma^{\scriptscriptstyle{OD}}_{o,k}}$${\Gamma^{\scriptscriptstyle{OD}}_{p,k}}$${\Gamma^{\scriptscriptstyle{OD}}_{o,l}}$${\Gamma^{\scriptscriptstyle{OD}}_{p,l}}$${\Gamma^{\scriptscriptstyle{BM}}}$

##  5 $\Gamma^{\scriptscriptstyle{BM}}_{r}$ and
$\Gamma^{\scriptscriptstyle{OD}}_{t,q}$ spaces

In this section, we study which spaces are
$\Gamma^{\scriptscriptstyle{BM}}_{r}$ and
$\Gamma^{\scriptscriptstyle{OD}}_{t,q}$ spaces. The importance of these spaces
is determined by the following proposition.

######  Proposition 4 (Proposition 15 [1]).

Let $X$ be a space and let
$\Gamma\in\\{\Gamma^{\scriptscriptstyle{BM}}_{r},\Gamma^{\scriptscriptstyle{OD}}_{t,q}\\}$,
where $r\in\\{o,p,f,k\\}$, $t\in\\{o,p\\}$ and $q\in\\{l,k\\}$.

  * (1)

If $X$ is nonmeager and $X$ is a $\Gamma$-space, then $X$ is
$\Gamma$-nonmeager.

  * (2)

If $X$ is a Baire space and $X$ is a $\Gamma$-space, then $X$ is a
$\Gamma$-Baire space.

######  Theorem 2 (Theorem 3 [1]).

  * (1)

$\Gamma^{\scriptscriptstyle{BM}}_{f}\subset\Gamma^{\scriptscriptstyle{BM}}_{k}\subset\Gamma^{\scriptscriptstyle{BM}}_{p}\subset\Gamma^{\scriptscriptstyle{BM}}_{o}$.

  * (2)

Let $r\in\\{o,p\\}$, $X\in\Gamma^{\scriptscriptstyle{BM}}_{r}$,
$Y\in\Gamma^{\scriptscriptstyle{BM}}_{k}$. Then $X\times
Y\in\Gamma^{\scriptscriptstyle{BM}}_{r}$.

  * (3)

Let $r\in\\{k,f\\}$, $X_{n}\in\Gamma^{\scriptscriptstyle{BM}}_{r}$ for
${n\in\omega}$. Then
$\prod_{n\in\omega}X_{n}\in\Gamma^{\scriptscriptstyle{BM}}_{r}$.

  * (4)

For $r\in\\{f,k,p,o\\}$, if $X\in\Gamma^{\scriptscriptstyle{BM}}_{r}$ and
$U\subset X$ are open subspaces, then
$U\in\Gamma^{\scriptscriptstyle{BM}}_{r}$.

  * (5)

For $r\in\\{f,k,p,o\\}$, if a space $X$ is locally
$\Gamma^{\scriptscriptstyle{BM}}_{r}$ (that is, every point has a neighborhood
$U\in\Gamma^{\scriptscriptstyle{BM}}_{r}$), then
$X\in\Gamma^{\scriptscriptstyle{BM}}_{r}$.

  * (6)

Let $t\in\\{f,k,p,o\\}$. If $X$ is a quasi-regular space and belongs to one of
the classes listed in the item $(\Gamma^{\scriptscriptstyle{BM}}_{t})$, then
$X$ is a $\Gamma^{\scriptscriptstyle{BM}}_{t}$-space.

    * $\mathrm{(\Gamma^{\scriptscriptstyle{BM}}_{f})}$

metrizable spaces, Moore spaces, developable space, semiregular
$\sigma$-spaces, and semiregular spaces with a countable network;

    * $\mathrm{(\Gamma^{\scriptscriptstyle{BM}}_{k})}$

compact spaces, $p$-spaces, semiregular strongly $\Sigma$-spaces;

    * $\mathrm{(\Gamma^{\scriptscriptstyle{BM}}_{p})}$

countably compact spaces, semiregular $\Sigma$-spaces, $w\Delta$-spaces;

    * $\mathrm{(\Gamma^{\scriptscriptstyle{BM}}_{o})}$

feebly compact spaces.

######  Theorem 3 (Theorem 4 [1]).

  * (1)

In the diagram below, the arrow $A\to B$ means that $A\subset B$.

${\Gamma^{\scriptscriptstyle{BM}}_{o}}$${\Gamma^{\scriptscriptstyle{BM}}_{k}}$${\Gamma^{\scriptscriptstyle{BM}}_{p}}$${\Gamma^{\scriptscriptstyle{OD}}_{o,k}}$${\Gamma^{\scriptscriptstyle{OD}}_{p,k}}$${\Gamma^{\scriptscriptstyle{OD}}_{o,l}}$${\Gamma^{\scriptscriptstyle{OD}}_{p,l}}$

  * (2)

Let $t\in\\{o,p\\}$, $X$ is a $\Gamma^{\scriptscriptstyle{OD}}_{t,k}$-space
and $Y$ is a $\Gamma^{\scriptscriptstyle{OD}}_{t,l}$-space. Then $X\times Y$
is a $\Gamma^{\scriptscriptstyle{OD}}_{t,l}$-space.

  * (3)

Let $D$ be an index set and let $X_{\delta}$ be an
$\Gamma^{\scriptscriptstyle{OD}}_{o,k}$-space for $\delta\in D$. Then
$\prod_{\delta\in D}X_{\delta}$ is a
$\Gamma^{\scriptscriptstyle{OD}}_{o,k}$-space.

  * (4)

Let $X_{n}$ be a $\Gamma^{\scriptscriptstyle{OD}}_{p,k}$-space for
${n\in\omega}$. Then $\prod_{{n\in\omega}}X_{n}$ is a
$\Gamma^{\scriptscriptstyle{OD}}_{p,k}$-space.

  * (5)

Let $t\in\\{o,p\\}$ and $q\in\\{l,k\\}$. If
$X\in\Gamma^{\scriptscriptstyle{OD}}_{t,q}$ and $U\subset X$ is an open
subspace, then $U\in\Gamma^{\scriptscriptstyle{OD}}_{t,q}$.

  * (6)

Let $t\in\\{o,p\\}$ and $q\in\\{l,k\\}$. If a space $X$ is locally
$\Gamma^{\scriptscriptstyle{OD}}_{t,q}$ (i.e., any point has a neighborhood
$U\in\Gamma^{\scriptscriptstyle{OD}}_{t,q}$), then
$X\in\Gamma^{\scriptscriptstyle{OD}}_{t,q}$.

  * (7)

Let $t\in\\{o,p\\}$ and $q\in\\{l,k\\}$. If $X$ is a quasi-regular space and
belongs to one of the classes listed in the item
$(\Gamma^{\scriptscriptstyle{OD}}_{t,q})$, then $X$ is a
$\Gamma^{\scriptscriptstyle{OD}}_{t,q}$-space.

    * $\mathrm{(\Gamma^{\scriptscriptstyle{OD}}_{p,k})}$

metrizable spaces, Moore spaces, developable space, semiregular
$\sigma$-spaces and semiregular spaces with a countable network, compact
spaces, $p$-spaces, semiregular strongly $\Sigma$-spaces;

    * $\mathrm{(\Gamma^{\scriptscriptstyle{OD}}_{p,l})}$

countably compact spaces, semiregular $\Sigma$-spaces, $w\Delta$-spaces;

    * $\mathrm{(\Gamma^{\scriptscriptstyle{OD}}_{o,l})}$

feebly compact spaces.

##  6 Playing against tactics

A strategy in a sequential game is called a tactic if the player takes into
account only the opponent’s previous move in his choice.

Let $(X,{\mathcal{T}})$ be a space and
${{\mathcal{T}}^{*}}={\mathcal{T}}\setminus\\{\varnothing\\}$, ${\mathcal{N}}$
be a $\pi$-net $X$. We denote

$\displaystyle T(X,{\mathcal{N}})$
$\displaystyle\coloneqq\\{\psi\in{\mathcal{N}}^{{\mathcal{T}}^{*}}\,:\,M\subset
U\text{ for }U\in{{\mathcal{T}}^{*}}\text{ and }M=\psi(U)\\},$ $\displaystyle
T_{BM}(X)$ $\displaystyle\coloneqq T(X,{{\mathcal{T}}^{*}}),$ $\displaystyle
T_{OD}(X,{\mathcal{N}})$ $\displaystyle\coloneqq
T(X,{{\mathcal{T}}^{*}})\times T(X,{\mathcal{N}}).$

Each $\varphi\in T_{BM}(X)$ uniquely corresponds to the tactics of player
$\beta$ in the games $BM(X,{\mathcal{V}})$ and $MB(X,{\mathcal{V}})$: on the
$n$-th move player $\beta$ selects $V_{n}=\varphi(U_{n})$. Each
$(\varphi,\psi)\in T_{OD}(X,{\mathcal{N}})$ uniquely corresponds to the
tactics of player $\beta$ in the games $OD(X,{\mathcal{N}},\mathcal{W})$ and
$DO(X,{\mathcal{N}},\mathcal{W})$: on the $n$th move, player $\beta$ chooses
$V_{n}=\varphi(U_{n})$ and $M_{n}=\psi(U_{n})$.

The modification of the games $BM$, $MB$, $OD$ and $DO$, in which the player
$\beta$ uses only tactics, will be denoted by $\widehat{BM}$, $\widehat{MB}$,
$\widehat{OD}$, $\widehat{DO}$, respectively.

We will assume that

  * 1.

$T_{BM}(X)$ is the set of strategies of player $\beta$ in the games
$\widehat{BM}$ and $\widehat{MB}$;

  * 2.

$T_{OD}(X,{\mathcal{N}})$ is the set of strategies of player $\beta$ in the
games $\widehat{OD}$ and $\widehat{DO}$.

In this section, we will give variants of definitions in which games are
replaced by games ‘‘with a hat’’, that is, those in which the player $\beta$
uses only tactics.

######  Definition 6.

Let $X$ be a space, ${\mathcal{V}}\subset{\mathfrak{V}}(X)$,
${\mathcal{V}}^{*}={\mathcal{V}}\cup{\mathfrak{V}}_{BM}^{*}(X)$,
${\mathcal{N}}$ be a $\pi$-net $X$, $\mathcal{W}\subset{\mathfrak{W}}(X)$ and
$\mathcal{W}^{*}=\mathcal{W}\cup{\mathfrak{W}}_{e}(X)$. We call a space $X$

  * 1.

$\Gamma^{\widehat{\scriptscriptstyle{BM}}}({\mathcal{V}})$-nonmeager if
$\widehat{MB}(X,{\mathcal{V}})$ is $\beta$-unfavorable;

  * 2.

$\Gamma^{\widehat{\scriptscriptstyle{BM}}}({\mathcal{V}})$-Baire if
$\widehat{BM}(X,{\mathcal{V}})$ is $\beta$-unfavorable;

  * 3.

$\Gamma^{\widehat{\scriptscriptstyle{BM}}}({\mathcal{V}})$-space if
$\widehat{BM}(X,{\mathcal{V}}^{*})$ is $\alpha$-favourable;

  * 4.

$\Gamma^{\widehat{\scriptscriptstyle{OD}}}({\mathcal{N}},\mathcal{W})$-nonmeager
if $\widehat{DO}(X,{\mathcal{N}},\mathcal{W})$ is $\beta$-unfavorable;

  * 5.

$\Gamma^{\widehat{\scriptscriptstyle{OD}}}({\mathcal{N}},\mathcal{W})$-Baire
if $\widehat{OD}(X,{\mathcal{N}},\mathcal{W})$ is $\beta$-unfavorable;

  * 6.

$\Gamma^{\widehat{\scriptscriptstyle{OD}}}({\mathcal{N}},\mathcal{W})$-space
if $\widehat{OD}(X,{\mathcal{N}},\mathcal{W}^{*})$ is $\alpha$-favourable.

######  Proposition 5.

Let ${\mathfrak{g}}$ be a game with two players $\alpha$ and $\beta$ and let
$T_{\beta}\subset S_{\beta}$ be some subset of the set of strategies
$S_{\beta}$ player $\beta$. We denote by $\hat{\mathfrak{g}}$ the game that
differs from the game ${\mathfrak{g}}$ in that player $\beta$ uses only
strategies from $T_{\beta}$. Then

  * (1)

if the game ${\mathfrak{g}}$ is $\alpha$-favourable, then the game
$\hat{\mathfrak{g}}$ is $\alpha$-favourable;

  * (2)

if the game $\hat{\mathfrak{g}}$ is $\beta$-unfavorable, then the game
${\mathfrak{g}}$ is $\beta$-unfavorable.

###### Proof.

(1) A strategy that is winning for $\alpha$ in the game ${\mathfrak{g}}$ will
also be winning in the game $\hat{\mathfrak{g}}$. (2) Let $s_{\beta}\in
T_{\beta}$ be a strategy for $\beta$ in the game $\hat{\mathfrak{g}}$. Since
$T_{\beta}\subset S_{\beta}$, we have $s_{\beta}\in S_{\beta}$. Since the game
$\hat{\mathfrak{g}}$ is $\beta$-unfavorable, player $\alpha$ has a strategy
that suppresses $s_{\alpha}$. ∎

Proposition 5 it implies the following assertion.

######  Proposition 6.

Let $X$ be a space, ${\mathcal{V}}\subset{\mathfrak{V}}(X)$,
${\mathcal{V}}^{*}={\mathcal{V}}\cup{\mathfrak{V}}_{BM}^{*}(X)$,
${\mathcal{N}}$ be a $\pi$-net $X$, $\mathcal{W}\subset{\mathfrak{W}}(X)$ and
$\mathcal{W}^{*}=\mathcal{W}\cup{\mathfrak{W}}_{e}(X)$. Then

  * 1.

if $X$ is
$\Gamma^{\widehat{\scriptscriptstyle{BM}}}({\mathcal{V}})$-nonmeager, then $X$
is $\Gamma^{\scriptscriptstyle{BM}}({\mathcal{V}})$-nonmeager;

  * 2.

if $X$ is $\Gamma^{\widehat{\scriptscriptstyle{BM}}}({\mathcal{V}})$-Baire,
then $X$ is $\Gamma^{\scriptscriptstyle{BM}}({\mathcal{V}})$-Baire;

  * 3.

if $X$ is a $\Gamma^{\scriptscriptstyle{BM}}({\mathcal{V}})$-space, then $X$
is a $\Gamma^{\widehat{\scriptscriptstyle{BM}}}({\mathcal{V}})$-space;

  * 4.

if $X$ is
$\Gamma^{\widehat{\scriptscriptstyle{OD}}}({\mathcal{N}},\mathcal{W})$-nonmeager,
then $X$ is
$\Gamma^{\scriptscriptstyle{OD}}({\mathcal{N}},\mathcal{W})$-nonmeager;

  * 5.

if $X$ is
$\Gamma^{\widehat{\scriptscriptstyle{OD}}}({\mathcal{N}},\mathcal{W})$-Baire,
then $X$ is
$\Gamma^{\scriptscriptstyle{OD}}({\mathcal{N}},\mathcal{W})$-Baire;

  * 6.

if $X$ is a
$\Gamma^{\scriptscriptstyle{OD}}({\mathcal{N}},\mathcal{W})$-space, then $X$
is a
$\Gamma^{\widehat{\scriptscriptstyle{OD}}}({\mathcal{N}},\mathcal{W})$-space.

By analogy with the definition of 3, we give the following definitions.

For $r\in\\{o,p,f,k\\}$ we define the games

$\displaystyle\widehat{BM}_{r}(X)$
$\displaystyle\coloneqq\widehat{BM}(X,{\mathcal{V}}),$
$\displaystyle\widehat{MB}_{r}(X)$
$\displaystyle\coloneqq\widehat{MB}(X,{\mathcal{V}}),$

where ${\mathcal{V}}={\mathfrak{V}}_{r}(X)$. For $t\in\\{o,p\\}$ and
$q\in\\{l,k\\}$ we define the games

$\displaystyle\widehat{OD}_{t,q}(X)$
$\displaystyle\coloneqq\widehat{OD}(X,\mathcal{N},\mathcal{W}),$
$\displaystyle\widehat{DO}_{t,q}(X)$
$\displaystyle\coloneqq\widehat{DO}(X,\mathcal{N},\mathcal{W})$

where $\mathcal{N}={\mathcal{N}}_{t}(X)$ and
$\mathcal{W}={\mathfrak{W}}_{q}(X)$.

######  Definition 7.

Let $X$ be a space, $r\in\\{o,p,f,k\\}$. Let us call a space $X$

  * 1.

$\Gamma^{\widehat{\scriptscriptstyle{BM}}}_{r}$-nonmeager if $X$ is
$(\beta,\widehat{MB}_{r})$-unfavourable;

  * 2.

$\Gamma^{\widehat{\scriptscriptstyle{BM}}}_{r}$-Baire if $X$ is
$(\beta,\widehat{BM}_{r})$-unfavourable.

Let $t\in\\{o,p\\}$ and $q\in\\{l,k\\}$. We call a space $X$

  * 1.

$\Gamma^{\widehat{\scriptscriptstyle{OD}}}_{t,q}$-nonmeager if $X$ is
$(\beta,\widehat{DO}_{t,q})$-unfavorable;

  * 2.

$\Gamma^{\widehat{\scriptscriptstyle{OD}}}_{t,q}$-Baire if $X$ is
$(\beta,\widehat{OD}_{t,q})$-unfavourable.

Proposition 6 implies the following assertion.

######  Proposition 7.

Let $X$ be a space, $r\in\\{o,p,f,k\\}$, $t\in\\{o,p\\}$ and $q\in\\{l,k\\}$.
Then

  * 1.

if $X$ is $\Gamma^{\widehat{\scriptscriptstyle{BM}}}_{r}$-nonmeager
($\Gamma^{\widehat{\scriptscriptstyle{BM}}}_{r}$-Baire), then $X$ is
$\Gamma^{\scriptscriptstyle{BM}}_{r}$-nonmeager
($\Gamma^{\scriptscriptstyle{BM}}_{r}$-Baire);

  * 2.

if $X$ is $\Gamma^{\widehat{\scriptscriptstyle{OD}}}_{t,q}$-nonmeager
($\Gamma^{\widehat{\scriptscriptstyle{OD}}}_{t,q}$-Baire), then $X$ is
$\Gamma^{\scriptscriptstyle{OD}}_{t,q}$\- nonmeager
($\Gamma^{\scriptscriptstyle{OD}}_{t,q}$-Baire).

The following proposition is easy to deduce from the definitions.

######  Proposition 8.

Let $X$ be a space.

  * 1.

Let ${\mathcal{V}}\subset{\mathfrak{V}}(X)$. A space $X$ is
$\Gamma^{\widehat{\scriptscriptstyle{BM}}}({\mathcal{V}})$-nonmeager
($\Gamma^{\widehat{\scriptscriptstyle{BM}}}({\mathcal{V}})$-Baire) if and only
if for any $\varphi\in T_{BM}(X)$ there exists
$(U_{n})_{n\in\omega}\in{\mathfrak{V}}(X)$ (for which $U_{0}=X$) such that

    * (a)

$V_{n}=\varphi(U_{n})$, $U_{n+1}\subset V_{n}$ for ${n\in\omega}$;

    * (b)

$(V_{n})_{n\in\omega}\in{\mathcal{V}}$.

  * 2.

Let ${\mathcal{N}}$ be a $\pi$-net $X$, $\mathcal{W}\subset{\mathfrak{W}}(X)$.
A space $X$ is
$\Gamma^{\widehat{\scriptscriptstyle{OD}}}({\mathcal{N}},\mathcal{W})$-nonmeager
($\Gamma^{\widehat{\scriptscriptstyle{OD}}}({\mathcal{N}},\mathcal{W})$-Baire)
if and only if for any $(\varphi,\psi)\in T_{OD}(X,{\mathcal{N}})$ there
exists $(U_{n})_{n\in\omega}\in{\mathfrak{V}}(X)$ (for which $U_{0}=X$) such
that

    * (a)

$V_{n}=\varphi(U_{n})$, $M_{n}=\psi(U_{n})$ and $U_{n+1}\subset V_{n}$ for
${n\in\omega}$;

    * (b)

$(V_{n},M_{n})_{n\in\omega}\in\mathcal{W}$.

##  7 Semi-neighborhoods of the diagonal

Let $X$ be a set and let $P,Q\subset X\times X$. We denote

$\displaystyle P(x)$ $\displaystyle\coloneqq\\{y\in X\,:\,(x,y)\in P\\},$
$\displaystyle P(M)$ $\displaystyle\coloneqq\\{y\in X\,:\,\text{ there exists
an }x\in M\text{ such that }(x,y)\in P\\}=\bigcup_{x\in M}P(x),$
$\displaystyle\left.P\right|_{M}$ $\displaystyle\coloneqq P\cap(M\times M),$
$\displaystyle P^{-1}$ $\displaystyle\coloneqq\\{(x,y)\,:\,(y,x)\in P\\},$
$\displaystyle P\circ Q$ $\displaystyle\coloneqq\\{(x,y)\,:\,\text{there is a
}z\in X\text{ such that }(x,z)\in P\text{ and }(z,y)\in Q\\}$

for $x\in X$ and $M\subset X$.

######  Definition 8.

Let $X$ be a space, $P\subset X\times X$. We call a set $P$ semiopen if every
$P(x)$ is open. We call $P$ a semi-neighbourhood of diagonal if
$x\in\operatorname{Int}P(x)$ for all $x\in X$. We denote by $\mathfrak{S}(X)$
the family of all semi-neighborhoods of the diagonal of the space $X$.

######  Remark 1.

In [2], in the definition of a semi-neighbourhood of a diagonal, it was
assumed that the set $P(x)$ is open, that is, $P$ was assumed to be semi-open.
The concept of a half-open semi-neighborhood of a diagonal essentially
coincides with the concept of neighbourhood assignment [10]: a mapping
$P:X\to{\mathcal{T}}$ is called a neighbourhood assignment if $x\in P(x)$ and
$P(x)$ is an open neighborhood of $x\in X$.

A set $P$ is a semi-neighborhood of the diagonal if and only if $P$ contains
some semi-open semi-neighborhood of the diagonal.

If $X$ is a topological space, then we denote

$\displaystyle\overline{P}^{v}$ $\displaystyle\coloneqq\\{(x,y)\in X\times
X\,:\,y\in\overline{P(x)}\\}.$

It is clear that $\overline{P}^{v}(x)=\overline{P(x)}$ for $x\in X$.

######  Proposition 9.

Let $X$ be a space, $P\subset X\times X$. Then

$\displaystyle\overline{P}(x)$
$\displaystyle=\bigcap\\{\overline{P(U)}\,:\,U\text{ is a neighborhood of
}x\\}$

for $x\in X$. Accordingly,

$\displaystyle\overline{P}$ $\displaystyle=\bigcap\\{\overline{Q\circ
P}^{v}\,:\,Q\in\mathfrak{S}(X)\\}.$

###### Proof.

Let $x\in X$. Then $(x,y)\in\overline{P}$ if and only if $y\in\overline{P(U)}$
for every neighborhood $U$ of $x$. ∎

On $\operatorname{Exp}(\mathfrak{S}(X))$ we introduce an order relation and an
equivalence relation related to it. For
${\mathcal{P}},\mathcal{Q}\subset\mathfrak{S}(X)$ we put

  * 1.

${\mathcal{P}}\preccurlyeq\mathcal{Q}$ if and only if for any
$Q\in\mathcal{Q}$ there exists a $P\in{\mathcal{P}}$ such that $P\subset Q$;

  * 2.

${\mathcal{P}}\sim\mathcal{Q}$ if and only if
${\mathcal{P}}\preccurlyeq\mathcal{Q}$ and
$\mathcal{Q}\preccurlyeq{\mathcal{P}}$.

Note that if ${\mathcal{P}}\supset\mathcal{Q}$, then
${\mathcal{P}}\preccurlyeq\mathcal{Q}$. For
${\mathcal{P}}\subset\mathfrak{S}(X)$ we denote

${\mathsf{P}}^{e}({\mathcal{P}})\coloneqq\\{R\in\mathfrak{S}(X)\,:\,P\subset
R,\text{ for some }P\in{\mathcal{P}}\\}.$

######  Proposition 10.

Let $X$ be a space. For ${\mathcal{P}},\mathcal{Q}\subset\mathfrak{S}(X)$

  * 1.

${\mathcal{P}}\preccurlyeq\mathcal{Q}$ if and only if
${\mathsf{P}}^{e}({\mathcal{P}})\supset{\mathsf{P}}^{e}(\mathcal{Q})$;

  * 2.

${\mathcal{P}}\sim\mathcal{Q}$ if and only if
${\mathsf{P}}^{e}({\mathcal{P}})={\mathsf{P}}^{e}(\mathcal{Q})$.

For ${\mathcal{P}}\subset\mathfrak{S}(X)$ we denote

$\displaystyle{\mathsf{P}}({\mathcal{P}})$
$\displaystyle\coloneqq{\mathcal{P}},$
$\displaystyle{\mathsf{P}}_{+}({\mathcal{P}})$
$\displaystyle\coloneqq\\{Q\circ P\,:\,Q\in\mathfrak{S}(X)\text{ and
}P\in{\mathcal{P}}\\},$ $\displaystyle{\mathsf{P}}^{v}({\mathcal{P}})$
$\displaystyle\coloneqq\\{\overline{P}^{v}\,:\,P\in{\mathcal{P}}\\},$
$\displaystyle{\mathsf{P}}^{c}({\mathcal{P}})$
$\displaystyle\coloneqq\\{\overline{P}\,:\,P\in{\mathcal{P}}\\},$
$\displaystyle{\mathsf{P}}_{+}^{c}({\mathcal{P}})$
$\displaystyle\coloneqq{\mathsf{P}}^{v}({\mathsf{P}}_{+}({\mathcal{P}})).$

Proposition 9 implies the following assertion.

######  Proposition 11.

Let $X$ be a space. For ${\mathcal{P}}\subset\mathfrak{S}(X)$

  * 1.

${\mathcal{P}}\preccurlyeq{\mathsf{P}}_{+}({\mathcal{P}})\preccurlyeq{\mathsf{P}}_{+}^{c}({\mathcal{P}})$,

  * 2.

${\mathcal{P}}\preccurlyeq{\mathsf{P}}^{v}({\mathcal{P}})\preccurlyeq{\mathsf{P}}^{c}({\mathcal{P}})\preccurlyeq{\mathsf{P}}_{+}^{c}({\mathcal{P}})$.

Let $\lambda$ be an ordinal and let $P_{\alpha}\subset X\times X$ for
$\alpha<\lambda$. By induction on $\beta<\gamma$ we define sets
${\mathsf{p}}_{\beta}(\mathcal{S}_{\beta}),{\mathsf{p}}^{c}_{\beta}(\mathcal{S}_{\beta})\subset
X\times X$, where $\mathcal{S}_{\beta}=(P_{\alpha})_{\alpha<\beta}$.

For $\beta=0$, we set

${\mathsf{p}}_{0}(\mathcal{S}_{0})\coloneqq{\mathsf{p}}_{0}^{c}(\mathcal{S}_{0})\coloneqq\\{\varnothing\\}.$

For $\beta=1$, we set

${\mathsf{p}}_{1}(\mathcal{S}_{1})\coloneqq
P_{0},\qquad{\mathsf{p}}_{1}^{c}(\mathcal{S}_{1})\coloneqq\overline{P_{0}}.$

For $1<\beta\leq\gamma$, we set

$\displaystyle{\mathsf{p}}_{\beta}(\mathcal{S}_{\beta})$
$\displaystyle\coloneqq\begin{cases}Q_{\beta}&\text{if $\beta$ is a limit
ordinal},\\\ Q_{\beta}\circ P_{\beta^{\prime}}&\text{if
}\beta=\beta^{\prime}+1,\end{cases}$
$\displaystyle{\mathsf{p}}^{c}_{\beta}(\mathcal{S}_{\beta})$
$\displaystyle\coloneqq\begin{cases}\overline{Q^{c}_{\beta}}&\text{if $\beta$
is a limit ordinal},\\\ \overline{Q^{c}_{\beta}\circ
P_{\beta^{\prime}}}&\text{if }\beta=\beta^{\prime}+1,\end{cases}$

where

$Q_{\beta}=\bigcup_{\alpha<\beta}{\mathsf{p}}_{\alpha}(\mathcal{S}_{\alpha})\qquad\text{and}\qquad
Q^{c}_{\beta}=\bigcup_{\alpha<\beta}{\mathsf{p}}^{c}_{\alpha}(\mathcal{S}_{\alpha}).$

For ${\mathcal{P}}\subset\mathfrak{S}(X)$ we denote

$\displaystyle{\mathsf{P}}_{\gamma}({\mathcal{P}})$
$\displaystyle\coloneqq\\{{\mathsf{p}}_{\gamma}(\mathcal{S})\,:\,\mathcal{S}\in{\mathcal{P}}^{\gamma}\\},$
$\displaystyle{\mathsf{P}}^{c}_{\gamma}({\mathcal{P}})$
$\displaystyle\coloneqq\\{{\mathsf{p}}^{c}_{\gamma}(\mathcal{S})\,:\,\mathcal{S}\in{\mathcal{P}}^{\gamma}\\},$
$\displaystyle\left.{\mathcal{P}}\right|_{Y}$
$\displaystyle\coloneqq\\{\left.P\right|_{Y}\,:\,P\in{\mathcal{P}}\\}$

for $Y\subset X$.

Proposition 11 and definitions imply the following assertion.

######  Proposition 12.

Let $X$ be a space, ${\mathcal{P}}\subset\mathfrak{S}(X)$, and
$1<\alpha<\beta$ ordinals. Then

$\displaystyle{\mathsf{P}}_{+}({\mathcal{P}})\preccurlyeq{\mathsf{P}}_{2}({\mathcal{P}})\preccurlyeq$
$\displaystyle{\mathsf{P}}_{\alpha}({\mathcal{P}})\preccurlyeq{\mathsf{P}}_{\beta}({\mathcal{P}})\preccurlyeq{\mathsf{P}}_{\beta}^{c}({\mathcal{P}}),$
$\displaystyle{\mathsf{P}}^{c}({\mathcal{P}})={\mathsf{P}}_{1}^{c}({\mathcal{P}})\preccurlyeq$
$\displaystyle{\mathsf{P}}_{\alpha}^{c}({\mathcal{P}})\preccurlyeq{\mathsf{P}}_{\beta}^{c}({\mathcal{P}}),$
$\displaystyle{\mathsf{P}}^{c}({\mathcal{P}})\preccurlyeq{\mathsf{P}}^{c}_{+}({\mathcal{P}})\preccurlyeq$
$\displaystyle{\mathsf{P}}_{2}^{c}({\mathcal{P}}).$

######  Proposition 13.

Let $X$ be a space, $Y\subset X$, ${\mathcal{P}}\subset\mathfrak{S}(X)$, and
$\gamma$ be an ordinal. If
$\tilde{\mathsf{P}}\in\\{{\mathsf{P}}^{v},{\mathsf{P}}^{c},{\mathsf{P}}_{+},{\mathsf{P}}^{c}_{+},{\mathsf{P}}_{\gamma},{\mathsf{P}}_{\gamma}^{c}\\}$,
then

$\tilde{\mathsf{P}}(\left.{\mathcal{P}}\right|_{Y})\preccurlyeq\left.\tilde{\mathsf{P}}({\mathcal{P}})\right|_{Y}.$

######  Proposition 14.

Let $X$ be a space, $\lambda$ be an ordinal, and $U\subset X$ be a nonempty
open subset of $X$. Then

$\displaystyle\left.{\mathsf{P}}_{\gamma}(\mathfrak{S}(X))\right|_{U}$
$\displaystyle\sim{\mathsf{P}}_{\gamma}(\mathfrak{S}(X)),$
$\displaystyle\left.{\mathsf{P}}^{c}_{\gamma}(\mathfrak{S}(X))\right|_{U}$
$\displaystyle\sim{\mathsf{P}}^{c}_{\gamma}(\mathfrak{S}(U)).$

######  Definition 9.

Let $(X,{\mathcal{T}})$ be a space,
${{\mathcal{T}}^{*}}={\mathcal{T}}\setminus\\{\varnothing\\}$ and
${\mathcal{P}}\subset\mathfrak{S}(X)$. We call the family ${\mathcal{P}}$
partible if the following condition holds:

  * $(P)$

Let $A$ be an index set, a family $\\{U_{\alpha}\,:\,\alpha\in
A\\}\subset{{\mathcal{T}}^{*}}$ be disjoint and $\\{P_{\alpha}\,:\,\alpha\in
A\\}\subset{\mathcal{P}}$. Then there are a $P\in{\mathcal{P}}$ and a family
of open sets $\\{V_{\alpha}\,:\,\alpha\in A\\}\subset{{\mathcal{T}}^{*}}$ such
that for every $\alpha\in A$ the following conditions are satisfied:

    * $(1)$

$V_{\alpha}\subset U_{\alpha}\subset\overline{V_{\alpha}}$;

    * $(2)$

$P(V_{\alpha})\subset U_{\alpha}$;

    * $(3)$

if $x\in V_{\alpha}$, then $P(x)\subset P_{\alpha}(x)$.

The definition implies the following assertion.

######  Proposition 15.

For any space $X$ the family $\mathfrak{S}(X)$ is partible.

######  Proposition 16.

Let $X$ be a space, ${\mathcal{P}},\mathcal{Q}\subset\mathfrak{S}(X)$ and
${\mathcal{R}}=\\{P\circ Q\,:\,P\in{\mathcal{P}}\text{ and
}Q\in\mathcal{Q}\\}$. If ${\mathcal{P}}$ and $\mathcal{Q}$ are partible
families, then ${\mathcal{R}}$ is a partible family.

###### Proof.

Let ${\mathcal{T}}$ be the topology of $X$, and let
${{\mathcal{T}}^{*}}={\mathcal{T}}\setminus\\{\varnothing\\}$.

Let $A$ be an index set, let $\\{U_{\alpha}\,:\,\alpha\in
A\\}\subset{{\mathcal{T}}^{*}}$ be a disjoint family, and let
$\\{R_{\alpha}\,:\,\alpha\in A\\}\subset{\mathcal{R}}$, where
$R_{\alpha}=P_{\alpha}\circ Q_{\alpha}$ for $\alpha\in A$. Since
${\mathcal{P}}$ is a partible family, there exists a $P\in{\mathcal{P}}$ and a
family of open sets $\\{W_{\alpha}\,:\,\alpha\in
A\\}\subset{{\mathcal{T}}^{*}}$ such that for every $\alpha\in A$ the
following conditions are met: $W_{\alpha}\subset
U_{\alpha}\subset\overline{W_{\alpha}}$; $P(W_{\alpha})\subset U_{\alpha}$; if
$x\in W_{\alpha}$, then $P(x)\subset P_{\alpha}(x)$. Since $\mathcal{Q}$ is a
partible family, there exists a $Q\in\mathcal{Q}$ and a family of open sets
$\\{V_{\alpha}\,:\,\alpha\in A\\}\subset{{\mathcal{T}}^{*}}$ such that for
every $\alpha\in A$ the following conditions are met: $V_{\alpha}\subset
W_{\alpha}\subset\overline{V_{\alpha}}$; $Q(V_{\alpha})\subset W_{\alpha}$; if
$x\in V_{\alpha}$, then $Q(x)\subset Q_{\alpha}(x)$.

Let $R=P\circ Q$. Then for each $\alpha\in A$ the following conditions are
satisfied: $V_{\alpha}\subset U_{\alpha}\subset\overline{V_{\alpha}}$;
$R(V_{\alpha})\subset U_{\alpha}$; if $x\in V_{\alpha}$, then $R(x)\subset
R_{\alpha}(x)$. ∎

######  Proposition 17.

Let $(X,{\mathcal{T}})$ be a quasi-regular space. For every open nonempty
$U\subset X$, there exists a disjoint family $\gamma$ of open subsets of
$X$such that

  * $(1)$

$\bigcup\gamma\subset U\subset\overline{\bigcup\gamma}$;

  * $(2)$

$\overline{V}\subset U$ for each $V\in\gamma$.

###### Proof.

Let ${\mathcal{T}}$ be the topology of the space $X$,
${{\mathcal{T}}^{*}}={\mathcal{T}}\setminus\\{\varnothing\\}$ and

$\nu=\\{V\in{{\mathcal{T}}^{*}}\,:\,\overline{V}\subset U\\}.$

For $\gamma$ any maximal disjoint subfamily of $\nu$ can be taken. ∎

######  Statement 1.

Let $(X,{\mathcal{T}})$ be a quasi-regular space and let
${{\mathcal{T}}^{*}}={\mathcal{T}}\setminus\\{\varnothing\\}$,
${\mathcal{P}}\subset\mathfrak{S}(X)$ be a partible family. Let $A$ be an
index set, $\\{U_{\alpha}\,:\,\alpha\in A\\}\subset{{\mathcal{T}}^{*}}$ be a
disjoint family, and $\\{P_{\alpha}\,:\,\alpha\in A\\}\subset{\mathcal{P}}$.
Then there are $P\in{\mathcal{P}}$ and a family of open sets
$\\{V_{\alpha}\,:\,\alpha\in A\\}\subset{{\mathcal{T}}^{*}}$ such that for
every $\alpha\in A$ the following conditions are satisfied:

  * $(1)$

$V_{\alpha}\subset U_{\alpha}\subset\overline{V_{\alpha}}$;

  * $(2)$

$\overline{P}(V_{\alpha})\subset U_{\alpha}$;

  * $(3)$

if $x\in V_{\alpha}$, then $P(x)\subset P_{\alpha}(x)$.

###### Proof.

We use Proposition 17 and for each $\alpha\in A$ choose a disjoint family
$\gamma_{\alpha}\subset{{\mathcal{T}}^{*}}$ such that
$\bigcup\gamma_{\alpha}\subset
U_{\alpha}\subset\overline{\bigcup\gamma_{\alpha}}$ and $\overline{U}\subset
U_{\alpha}$ for each $U\in\gamma_{\alpha}$. We denote $\alpha(U)=\alpha$ and
$P_{U}=P_{\alpha}$ for $U\in\gamma_{\alpha}$. We put
$\gamma=\bigcup_{\alpha\in A}\gamma_{\alpha}$. Since ${\mathcal{P}}$ is a
partible family, there exists a $P\in{\mathcal{P}}$ and a family of open sets
$\\{V_{U}\,:\,U\in\gamma\\}\subset{{\mathcal{T}}^{*}}$ such that for every
$U\in\gamma$ the following conditions are met:

  * 1.

$V_{U}\subset U\subset\overline{V_{U}}$;

  * 2.

$P(V_{U})\subset U$;

  * 3.

if $x\in V_{U}$ then $P(x)\subset P_{U}(x)$.

Let $V_{\alpha}=\bigcup\\{V_{U}\,:\,U\in\gamma_{\alpha}\\}$ for $\alpha\in A$.
By construction, conditions (1) and (3) are satisfied. Let us check (2). Let
$x\in V_{\alpha}$. Then $x\in V_{U}$ for some $U\in\gamma_{\alpha}$. We obtain

$P(V_{U})\subset\overline{P(V_{U})}=\overline{U}\subset U_{\alpha}.$

Proposition 9 implies that $\overline{P}(x)\subset\overline{P(V_{U})}\subset
U_{\alpha}$. ∎

Propositions 15, 16 and Statement 1 imply the following assertion.

######  Proposition 18.

Let $X$ be a space and let ${\mathcal{P}}\subset\mathfrak{S}(X)$ be a partible
family.

  * 1.

The families ${\mathsf{P}}_{+}({\mathcal{P}})$ and
${\mathsf{P}}_{n}({\mathcal{P}})$ for $n<\omega$ are partible.

  * 2.

If $X$ is a quasi-regular space, then the families
${\mathsf{P}}^{v}({\mathcal{P}})$, ${\mathsf{P}}^{c}({\mathcal{P}})$,
${\mathsf{P}}_{+}^{c}({\mathcal{P}})$ and
${\mathsf{P}}_{n}^{c}({\mathcal{P}})$ for $n<\omega$ are partible.

######  Definition 10.

Let $X$ be a space and ${\mathcal{P}}\subset\mathfrak{S}(X)$. We denote

$\displaystyle\operatorname{Aut_{{\mathcal{P}}}}(X)\coloneqq\\{f\in\operatorname{Aut}(X):\
$ $\displaystyle(f\times f)(P)\in{\mathcal{P}}\text{ and}$
$\displaystyle(f^{-1}\times f^{-1})(P)\in{\mathcal{P}}$
$\displaystyle\text{for all }P\in{\mathcal{P}}\\}.$

We call a space $X$ ${\mathcal{P}}$-homogeneous if for any $x,y\in X$ there
exists an $f\in\operatorname{Aut_{{\mathcal{P}}}}(X)$ such that $f(x)=y$ .

The following proposition is a direct consequence of the definitions.

######  Proposition 19.

Let $X$ be a space, ${\mathcal{P}}\subset\mathfrak{S}(X)$, $\alpha$ be an
ordinal and

$\mathcal{Q}\in\\{{\mathsf{P}}^{e}({\mathcal{P}}),{\mathsf{P}}^{v}({\mathcal{P}}),{\mathsf{P}}^{c}({\mathcal{P}}),{\mathsf{P}}_{+}({\mathcal{P}}),{\mathsf{P}}^{c}_{+}({\mathcal{P}}),{\mathsf{P}}_{\alpha}({\mathcal{P}}),{\mathsf{P}}_{\alpha}^{c}({\mathcal{P}})\\}.$

Then

  * 1.

$\operatorname{Aut_{{\mathcal{P}}}}(X)\subset\operatorname{Aut_{\mathcal{Q}}}(X)$;

  * 2.

if $X$ is ${\mathcal{P}}$-homogeneous, then $X$ is $\mathcal{Q}$-homogeneous.

######  Proposition 20.

Let $X$ be a space, ${\mathcal{P}}=\mathfrak{S}(X)$, $\alpha$ be an ordinal,

$\mathcal{Q}\in\\{{\mathcal{P}},{\mathsf{P}}^{e}({\mathcal{P}}),{\mathsf{P}}^{v}({\mathcal{P}}),{\mathsf{P}}^{c}({\mathcal{P}}),{\mathsf{P}}_{+}({\mathcal{P}}),{\mathsf{P}}^{c}_{+}({\mathcal{P}}),{\mathsf{P}}_{\alpha}({\mathcal{P}}),{\mathsf{P}}_{\alpha}^{c}({\mathcal{P}})\\}.$

Then

  * 1.

$\operatorname{Aut}(X)=\operatorname{Aut_{\mathcal{Q}}}(X)$;

  * 2.

if $X$ is homogeneous, then $X$ is $\mathcal{Q}$-homogeneous.

##  8 Normal square functors

A functor ${\mathfrak{Q}}$ that associates a topological space $X$ with a
family ${\mathfrak{Q}}(X)\subset\operatorname{{Exp}}_{*}(X\times X)$ of
nonempty subsets of $X\times X$ is called a square functor. A square functor
${\mathfrak{Q}}$ is called a normal square functor (NSF) if the following
conditions are satisfied:

  * $(Q_{1})$

if $f:X\to Y$ is a homeomorphism, then

${\mathfrak{Q}}(Y)=\\{(f\times f)(P)\,:\,P\in{\mathfrak{Q}}(X)\\};$

  * $(Q_{2})$

if $U$ is an open nonempty subset of $X$, then

${\mathfrak{Q}}(U)=\left.{\mathfrak{Q}}(X)\right|_{U};$

  * $(Q_{3})$

if $S\in{\mathfrak{Q}}(X)$, $S\subset Q\subset X\times X$, then
$Q\in{\mathfrak{Q}}(X)$;

  * $(Q_{4})$

if $S\in{\mathfrak{Q}}(X)$, then $\overline{S}=X\times X$.

We introduce an order relation on normal square functors. For NSFs
${\mathfrak{Q}}$ and ${\mathfrak{R}}$
${\mathfrak{Q}}\preccurlyeq{\mathfrak{R}}$ is true if and only if
${\mathfrak{Q}}(X)\supset{\mathfrak{R}}(X)$ for any space $X$.

Let us define the NSFs which we will use. Let $k\in\\{d,h,v,s,a\\}$, $X$ be a
space and $S\in\operatorname{{Exp}}_{*}(X\times X)$. Then
$S\in{\mathfrak{Q}}_{k}(X)$ if and only if the corresponding condition
$(L_{k})$ holds:

  * $(L_{d})$

$\overline{S}=X\times X$;

  * $(L_{v})$

$\overline{S(x)}=X$ for any $x\in X$;

  * $(L_{h})$

$\overline{S^{-1}(x)}=X$ for any $x\in X$;

  * $(L_{s})$

$M\times X\subset S$ for some $M\subset X=\overline{M}$;

  * $(L_{a})$

$S=X$.

The definition imply the following assertion.

######  Proposition 21.

Let $k\in\\{d,h,v,s,a\\}$, $X$ be a space and
$S\in\operatorname{{Exp}}_{*}(X\times X)$. Then $S\in{\mathfrak{Q}}_{k}(X)$ if
and only if for any $x\in X$ and any neighborhood $U\subset X$ of $x$ the
corresponding condition $(L^{\prime}_{k})$ is satisfied:

  * $(L^{\prime}_{d})$

$\overline{S(U)}=X$;

  * $(L^{\prime}_{v})$

$\overline{S(x)}=X$;

  * $(L^{\prime}_{h})$

$S(U)=X$;

  * $(L^{\prime}_{s})$

$S(z)=X$ for some $z\in U$;

  * $(L^{\prime}_{a})$

$S(x)=X$.

######  Proposition 22.

For any NSF ${\mathfrak{Q}}$

$\displaystyle{\mathfrak{Q}}_{d}\preccurlyeq{\mathfrak{Q}}\preccurlyeq{\mathfrak{Q}}_{a},$
$\displaystyle{\mathfrak{Q}}_{h}\preccurlyeq{\mathfrak{Q}}_{s}.$

We introduce an increment operation on NSF: given an NSF ${\mathfrak{Q}}$, we
define an NSF ${\mathfrak{Q}}^{+}$. Let $X$ be a space and let
$Q\in\operatorname{{Exp}}_{*}(X\times X)$. Then $Q\in{\mathfrak{Q}}^{+}(X)$ if
and only if $P\circ S\subset Q$ for some $P\in\mathfrak{S}(X)$ and
$S\in{\mathfrak{Q}}(X)$.

######  Proposition 23.

$\displaystyle{\mathfrak{Q}}_{v}\preccurlyeq{\mathfrak{Q}}^{+}_{d}$
$\displaystyle{\mathfrak{Q}}_{a}={\mathfrak{Q}}^{+}_{s}={\mathfrak{Q}}^{+}_{h}$

##  9 $\Delta({\mathcal{P}};{\mathfrak{Q}})$-Baire spaces

######  Definition 11.

Let ${\mathfrak{Q}}$ be a normal square functor, $(X,{\mathcal{T}})$ a space,
${{\mathcal{T}}^{*}}={\mathcal{T}}\setminus\\{\varnothing\\}$,
${\mathcal{P}}\subset\mathfrak{S}(X)$. We call $X$

  * 1.

a $\Delta({\mathcal{P}};{\mathfrak{Q}})$-nonmeager
($\Delta({\mathcal{P}};{\mathfrak{Q}})$-nonmeager) space if for any
$P\in{\mathcal{P}}$ there exists $V\in{{\mathcal{T}}^{*}}$ such that
$\left.P\right|_{V}\in{\mathfrak{Q}}(V)$;

  * 2.

$\Delta({\mathcal{P}};{\mathfrak{Q}})$-Baire space
($\Delta({\mathcal{P}};{\mathfrak{Q}})$-Baire space) if for any
$P\in{\mathcal{P}}$ and any $U\in{{\mathcal{T}}^{*}}$ there exists an open
nonempty $V\subset U$ such that $\left.P\right|_{V}\in{\mathfrak{Q}}(V)$.

The definitions imply the following assertion.

######  Proposition 24.

Let $X$ be a space, ${\mathfrak{Q}}$, ${\mathfrak{R}}$ be NSF and
${\mathcal{P}},{\mathcal{R}}\subset\mathfrak{S}(X)$. Suppose that
${\mathfrak{Q}}\succcurlyeq{\mathfrak{R}}$ and
${\mathcal{P}}\preccurlyeq{\mathcal{R}}$.

  * 1.

If $X$ is a $\Delta({\mathcal{P}};{\mathfrak{Q}})$-Baire space, then $X$ is a
$\Delta({\mathcal{R}};{\mathfrak{R}})$-Baire space.

  * 2.

If $X$ is a $\Delta({\mathcal{P}};{\mathfrak{Q}})$-nonmeager space, then $X$
is a $\Delta({\mathcal{R}};{\mathfrak{R}})$-nonmeager space.

For $P\subset X\times X$ we denote

$\displaystyle\mathcal{W}(P,{\mathfrak{Q}})$
$\displaystyle\coloneqq\\{V\in{{\mathcal{T}}^{*}}:\left.P\right|_{V}\in{\mathfrak{Q}}(V)\\},$
$\displaystyle W(P,{\mathfrak{Q}})$
$\displaystyle\coloneqq\bigcup\mathcal{W}(P,{\mathfrak{Q}}).$

For ${\mathcal{P}}\subset\mathfrak{S}(X)$ we denote

$\displaystyle{\mathcal{B}}({\mathcal{P}},{\mathfrak{Q}})$
$\displaystyle\coloneqq\\{V\in{{\mathcal{T}}^{*}}:\text{$V$ is
$\Delta(\left.{\mathcal{P}}\right|_{V};{\mathfrak{Q}})$-Baire}\\},$
$\displaystyle B({\mathcal{P}},{\mathfrak{Q}})$
$\displaystyle\coloneqq\bigcup{\mathcal{B}}({\mathcal{P}},{\mathfrak{Q}}).$

The following three assertions are verified directly.

######  Statement 2.

$B({\mathcal{P}},{\mathfrak{Q}})\subset\overline{W(P,{\mathfrak{Q}})}$ for
$P\in{\mathcal{P}}$.

######  Statement 3.

For any $P\in{\mathcal{P}}$ a space $X$ is

  * 1.

$\Delta({\mathcal{P}};{\mathfrak{Q}})$-nonmeager space if and only if
$W(P,{\mathfrak{Q}})\neq\varnothing$,

  * 2.

$\Delta({\mathcal{P}};{\mathfrak{Q}})$-Baire if and only if
$\overline{W(P,{\mathfrak{Q}})}=X$.

######  Statement 4.

$W(P,{\mathfrak{Q}})\cap V=W(\left.P\right|_{V},{\mathfrak{Q}})$ for
$V\in{{\mathcal{T}}^{*}}$.

Statement 3 implies the following proposition.

######  Proposition 25.

Let $X$ be a space, ${\mathfrak{Q}}$ be an NSF and
${\mathcal{P}}\subset\mathfrak{S}(X)$. If $X$ is
$\Delta({\mathcal{P}};{\mathfrak{Q}})$-Baire, then $X$ is
$\Delta({\mathcal{P}};{\mathfrak{Q}})$-nonmeager.

Statements 3 and 4 imply the following proposition.

######  Proposition 26.

Let $X$ be a space, ${\mathfrak{Q}}$ be an NSF and
${\mathcal{P}}\subset\mathfrak{S}(X)$.

  * (1)

A space $X$ is $\Delta({\mathcal{P}};{\mathfrak{Q}})$-Baire if and only if $U$
is a $\Delta(\left.{\mathcal{P}}\right|_{U};{\mathfrak{Q}})$-nonmeager space
for any nonempty open $U\subset X$.

  * (2)

If $U\subset X$ is a nonempty open space and $U$ is a
$\Delta(\left.{\mathcal{P}}\right|_{U};{\mathfrak{Q}})$-nonmeager space, then
$X$ is $\Delta({\mathcal{P}};{\mathfrak{Q}})$-nonmeager.

######  Statement 5.

If $P,Q\in\mathfrak{S}(X)$, then $W(Q,{\mathfrak{Q}})\subset W(P\circ
Q,{\mathfrak{Q}}^{+})$.

###### Proof.

It suffices to check $\mathcal{W}(Q,{\mathfrak{Q}})\subset\mathcal{W}(P\circ
Q,{\mathfrak{Q}}^{+})$. Let $V\in\mathcal{W}(Q,{\mathfrak{Q}})$. Then
$\left.Q\right|_{V}\in{\mathfrak{Q}}(V)$ and hence
$\left.P\right|_{V}\circ\left.Q\right|_{V}\in{\mathfrak{Q}}^{+}(X)$. Since
$\left.P\right|_{V}\circ\left.Q\right|_{V}\subset\left.(P\circ Q)\right|_{V}$,
it follows that $\left.(P\circ Q)\right|_{V}\in{\mathfrak{Q}}^{+}(V)$. ∎

Statements 3 and 5 imply the following proposition.

######  Proposition 27.

Let $X$ be a space, ${\mathfrak{Q}}$ be an NSF and
${\mathcal{P}}\subset\mathfrak{S}(X)$.

  * (1)

If $X$ is a $\Delta({\mathcal{P}};{\mathfrak{Q}})$-nonmeager space, then $X$
is $\Delta({\mathsf{P}}_{+}({\mathcal{P}});{\mathfrak{Q}}^{+})$-nonmeager
space.

  * (2)

If $X$ is a $\Delta({\mathcal{P}};{\mathfrak{Q}})$-Baire space, then $X$ is a
$\Delta({\mathsf{P}}_{+}({\mathcal{P}});{\mathfrak{Q}}^{+})$-Baire space
space.

######  Statement 6.

If $P\in\mathfrak{S}(X)$, then $W(P,{\mathfrak{Q}}_{d})\subset
W(\overline{P},{\mathfrak{Q}}_{a})$.

###### Proof.

It suffices to check
$\mathcal{W}(P,{\mathfrak{Q}}_{d})\subset\mathcal{W}(\overline{P},{\mathfrak{Q}}_{a})$.
Let $V\in\mathcal{W}(P,{\mathfrak{Q}}_{d})$. Then
$\left.P\right|_{V}\in{\mathfrak{Q}}_{d}(V)$ and hence $V\times
V\subset\overline{\left.P\right|_{V}}\subset\overline{P}$. ∎

Statements 3 and 6 imply the following proposition.

######  Proposition 28.

Let $X$ be a space and let ${\mathcal{P}}\subset\mathfrak{S}(X)$.

  * (1)

If $X$ is a $\Delta({\mathcal{P}};{\mathfrak{Q}}_{d})$-nonmeager space, then
$X$ is a
$\Delta({\mathsf{P}}^{c}({\mathcal{P}});{\mathfrak{Q}}_{a})$-nonmeager space.

  * (2)

If $X$ is a $\Delta({\mathcal{P}};{\mathfrak{Q}}_{d})$-Baire space, then $X$
is a $\Delta({\mathsf{P}}^{c}({\mathcal{P}});{\mathfrak{Q}}_{a})$-Baire space.

Proposition 28 implies the following assertion.

######  Proposition 29.

Let $X$ be a space, ${\mathcal{P}}\subset\mathfrak{S}(X)$,
${\mathsf{P}}^{c}({\mathcal{P}})={\mathcal{P}}$ and ${\mathfrak{Q}}$ be an
NSF. Then

  * (1)

$X$ is a $\Delta({\mathcal{P}};{\mathfrak{Q}})$-nonmeager space if and only if
$X$ is a $\Delta({\mathcal{P}};{\mathfrak{Q}}_{a})$-nonmeager space;

  * (2)

$X$ is a $\Delta({\mathcal{P}};{\mathfrak{Q}})$-Baire space if and only if $X$
is a $\Delta({\mathcal{P}};{\mathfrak{Q}}_{a})$-Baire space.

Propositions 11, 12, 22, 23, 24, 27, 28 and 29 imply the following assertion.

######  Proposition 30.

Let $X$ be a space and let ${\mathcal{P}}\subset\mathfrak{S}(X)$. In the
diagrams below, the arrow

$(F)_{k}\to(G)_{l}$

means that
$F,G:\operatorname{Exp}(\mathfrak{S}(X))\to\operatorname{Exp}(\mathfrak{S}(X))$
are mappings, $k,l\in\\{d,v,h,s,a\\}$ and the following conditions hold:

  * (1)

If $X$ is a $\Delta(F({\mathcal{P}});{\mathfrak{Q}}_{k})$-nonmeager space,
then $X$ is $\Delta(G({\mathcal{P}});{\mathfrak{Q}}_{l})$-nonmeager space and

  * (2)

if $X$ is a $\Delta(F({\mathcal{P}});{\mathfrak{Q}}_{k})$-Baire space, then
$X$ is $\Delta(G({\mathcal{P}});{\mathfrak{Q}}_{l})$-Baire space.

${({\mathsf{P}}^{c})_{v}}$${({\mathsf{P}}^{c})_{a}}$${({\mathsf{P}}^{c})_{s}}$${({\mathsf{P}}^{c})_{h}}$${({\mathsf{P}}^{c})_{d}}$

${({\mathsf{P}})_{a}}$${({\mathsf{P}})_{s}}$${({\mathsf{P}})_{h}}$${({\mathsf{P}})_{v}}$${({\mathsf{P}})_{d}}$${({\mathsf{P}}_{+})_{a}}$${({\mathsf{P}}_{+})_{v}}$${({\mathsf{P}}_{+}^{c})_{a}}$${({\mathsf{P}}^{c})_{a}}$${({\mathsf{P}}_{2})_{a}}$${({\mathsf{P}}_{2})_{s}}$${({\mathsf{P}}_{2})_{h}}$${({\mathsf{P}}_{1}^{c})_{a}}$${({\mathsf{P}}_{2})_{v}}$${({\mathsf{P}}_{2})_{d}}$${({\mathsf{P}}_{2}^{c})_{a}}$

Let $1<\alpha<\beta$ be ordinals.

${({\mathsf{P}}_{\alpha}^{c})_{v}}$${({\mathsf{P}}_{\alpha}^{c})_{a}}$${({\mathsf{P}}_{\alpha}^{c})_{s}}$${({\mathsf{P}}_{\alpha}^{c})_{h}}$${({\mathsf{P}}_{\alpha}^{c})_{d}}$

${({\mathsf{P}}_{\alpha})_{a}}$${({\mathsf{P}}_{\alpha})_{s}}$${({\mathsf{P}}_{\alpha})_{h}}$${({\mathsf{P}}_{\alpha})_{v}}$${({\mathsf{P}}_{\alpha})_{d}}$${({\mathsf{P}}_{\beta})_{a}}$${({\mathsf{P}}_{\beta})_{s}}$${({\mathsf{P}}_{\beta})_{h}}$${({\mathsf{P}}_{\alpha}^{c})_{a}}$${({\mathsf{P}}_{\beta})_{v}}$${({\mathsf{P}}_{\beta})_{d}}$${({\mathsf{P}}_{\beta}^{c})_{a}}$

The following two assertions are easy to verify.

######  Statement 7.

Let $X$ be a space, ${\mathfrak{Q}}$ be an NSF and $P\in\mathfrak{S}(X)$. If
$f\in\operatorname{Aut}(X)$, then

$f(W(P,{\mathfrak{Q}}))=W((f\times f)(P),{\mathfrak{Q}}).$

######  Statement 8.

Let $X$ be a space, ${\mathfrak{Q}}$ be an NSF and
${\mathcal{P}}\subset\mathfrak{S}(X)$. If
$f\in\operatorname{Aut_{{\mathcal{P}}}}(X)$ then

$f(B({\mathcal{P}},{\mathfrak{Q}}))=B({\mathcal{P}},{\mathfrak{Q}}).$

######  Statement 9.

Let $X$ be a space, ${\mathcal{P}}\subset\mathfrak{S}(X)$ and ${\mathfrak{Q}}$
be an NSF. If $X$ is a ${\mathcal{P}}$-homogeneous space and
$B({\mathcal{P}},{\mathfrak{Q}})\neq\varnothing$, then $X$ is
$\Delta({\mathcal{P}};{\mathfrak{Q}})$-Baire.

###### Proof.

Let $U=B({\mathcal{P}},{\mathfrak{Q}})$. Let us show that $U=X$. Take an
arbitrary $y\in X$. Let $x\in U$. Since the space $X$ is
${\mathcal{P}}$-homogeneous, there exists an
$f\in\operatorname{Aut_{{\mathcal{P}}}}(X)$ such that $f(x)=y$. From Statement
8 it follows that $f(U)=U$. Hence $y\in U$. Proved $U=X$.

Let $P\in{\mathcal{P}}$. It follows from Statement 2 that
$U\subset\overline{W(P,{\mathfrak{Q}})}$. Hence
$\overline{W(P,{\mathfrak{Q}})}=X$. It follows from Statement 3 that $X$ is
$\Delta({\mathcal{P}};{\mathfrak{Q}})$-Baire. ∎

######  Statement 10.

Let $X$ be a space, ${\mathcal{P}}\subset\mathfrak{S}(X)$ be a partible family
and ${\mathfrak{Q}}$ be an NSF. The space $X$ is
$\Delta({\mathcal{P}};{\mathfrak{Q}})$-nonmeager if and only if
$B({\mathcal{P}},{\mathfrak{Q}})\neq\varnothing$.

###### Proof.

Let ${\mathcal{T}}$ be the topology of $X$ and let
${{\mathcal{T}}^{*}}={\mathcal{T}}\setminus\\{\varnothing\\}$. Propositions 25
and 26 imply that if $U$ is a
$\Delta(\left.{\mathcal{P}}\right|_{U};{\mathfrak{Q}})$-Baire space for some
$U\in{{\mathcal{T}}^{*}}$, then $X$ is
$\Delta({\mathcal{P}};{\mathfrak{Q}})$-nonmeager.

Let us prove the converse assertion. Assume that, on the contrary,
$B({\mathcal{P}},{\mathfrak{Q}})=\varnothing$. Then $U$ is not a
$\Delta(\left.{\mathcal{P}}\right|_{U};{\mathfrak{Q}})$-Baire space for any
$U\in{{\mathcal{T}}^{*}}$. Proposition 26 implies that $V$ is not
$\Delta(\left.{\mathcal{P}}\right|_{V};{\mathfrak{Q}})$-nonmeager for some
$V\in{{\mathcal{T}}^{*}}$, $V\subset U$. We set

$\nu=\\{V\in{{\mathcal{T}}^{*}}\,:\,V\text{ is not
$\Delta(\left.{\mathcal{P}}\right|_{V};{\mathfrak{Q}})$-nonmeager}\\}.$

The family $\nu$ is a $\pi$-base of $X$. Let $\gamma\subset\nu$ be a maximal
disjoint subfamily of the family $\nu$. Then $\overline{\bigcup\gamma}=X$. It
follows from Statement 3 that for each $V\in\gamma$ there exists a
$P_{V}\in{\mathcal{P}}$ for which

$W(P_{V},{\mathfrak{Q}})\cap
V=W(\left.P_{V}\right|_{V},{\mathfrak{Q}})=\varnothing.$

Since ${\mathcal{P}}$ is a partible family, there exist $P\in{\mathcal{P}}$
and $W_{V}\in{{\mathcal{T}}^{*}}$ for $V\in\gamma$ such that the following
conditions hold:

  * 1.

$W_{V}\subset V\subset\overline{W_{V}}$;

  * 2.

$P(W_{V})\subset V$;

  * 3.

if $x\in W_{V}$ then $P(x)\subset P_{V}(x)$.

We obtain $W(P,{\mathfrak{Q}})\cap W_{V}\subset W(P_{V},{\mathfrak{Q}})\cap
W_{V}$ and $W(P,{\mathfrak{Q}})\cap V=\varnothing$ for $V\in\gamma$. Hence
$W(P,{\mathfrak{Q}})=\varnothing$. Statement 3 implies that $X$ is not
$\Delta({\mathcal{P}};{\mathfrak{Q}})$-nonmeager. This contradiction proves
the statement. ∎

######  Proposition 31.

Let $X$ be a space, ${\mathcal{P}}\subset\mathfrak{S}(X)$ be a partible family
and ${\mathfrak{Q}}$ be an NSF. Assume that $X$ is a
${\mathcal{P}}$-homogeneous space. Then the following conditions are
equivalent:

  * (1)

The space $X$ is $\Delta({\mathcal{P}};{\mathfrak{Q}})$-Baire.

  * (2)

The space $X$ is $\Delta({\mathcal{P}};{\mathfrak{Q}})$-nonmeager.

###### Proof.

The implication $(1)\implies(2)$ follows from Proposition 25. Let us prove
$(2)\implies(1)$. It follows from Statement 10 that
$B({\mathcal{P}},{\mathfrak{Q}})\neq\varnothing$. It follows from Statement 9
that $X$ is $\Delta({\mathcal{P}};{\mathfrak{Q}})$-Baire. ∎

Proposition 21 implies the following assertion.

######  Proposition 32.

Let $X$ be a space, ${\mathcal{P}}\subset\mathfrak{S}(X)$, $Y\subset
X=\overline{Y}$, and $\mathcal{Q}=\left.{\mathcal{P}}\right|_{Y}$.

  * (1)

If $Y$ is $\Delta(\mathcal{Q};{\mathfrak{Q}}_{d})$-nonmeager, then $X$ is
$\Delta({\mathcal{P}};{\mathfrak{Q}}_{d})$-nonmeager.

  * (2)

Let $k\in\\{h,s\\}$. If $Y$ is
$\Delta(\mathcal{Q};{\mathfrak{Q}}_{k})$-nonmeager, then $X$ is
$\Delta({\mathsf{P}}^{v}({\mathcal{P}});{\mathfrak{Q}}_{k})$-nonmeager.

##  10 $\Delta({\mathfrak{Q}})$-Baire spaces

######  Definition 12.

Let ${\mathfrak{Q}}$ be a normal square functor, $X$ be a space, $\gamma$ be
an ordinal, and ${\mathcal{P}}=\mathfrak{S}(X)$. Let
${\mathsf{P}}^{k}_{l}:\operatorname{Exp}(\mathfrak{S}(X))\to\operatorname{Exp}(\mathfrak{S}(X))$
be one of the mappings considered in Section 7:

${\mathsf{P}}^{k}_{l}\in\\{{\mathsf{P}},{\mathsf{P}}^{e},{\mathsf{P}}^{v},{\mathsf{P}}^{c},{\mathsf{P}}_{+},{\mathsf{P}}_{+}^{c},{\mathsf{P}}_{\gamma},{\mathsf{P}}_{\gamma}^{c}\\}.$

We say that $X$ is a $\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{k}$\hss}\,\,}}}{{{}^{l}}}{\operatorname{\Delta}}({\mathfrak{Q}})$-Baire
($\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{k}$\hss}\,\,}}}{{{}^{l}}}{\operatorname{\Delta}}({\mathfrak{Q}})$-nonmeager)
space if $X$ is a
$\Delta({\mathsf{P}}^{k}_{l}(\mathfrak{S}(X));{\mathfrak{Q}})$-Baire
($\Delta({\mathsf{P}}^{k}_{l}(\mathfrak{S}(X));{\mathfrak{Q}})$-nonmeager)
space. For

${\tilde{\Delta}}\in\\{\Delta,\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{e}$\hss}\,\,}}}{{}}{\operatorname{\Delta}},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{v}$\hss}\,\,}}}{{}}{\operatorname{\Delta}},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{c}$\hss}\,\,}}}{{}}{\operatorname{\Delta}},\Delta^{+},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{c}$\hss}\,\,}}}{{{}^{+}}}{\operatorname{\Delta}},\Delta^{\gamma},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{c}$\hss}\,\,}}}{{{}^{\gamma}}}{\operatorname{\Delta}}\\}$

we have defined ${\tilde{\Delta}}({\mathfrak{Q}})$-Baire
(${\tilde{\Delta}}({\mathfrak{Q}})$-nonmeager) spaces. For

$k\in\\{d,h,v,s,a\\}$

we say that $X$ is a ${{\tilde{\Delta}}}_{k}$-Baire
(${{\tilde{\Delta}}}_{k}$-nonmeager) space if $X$ is
${\tilde{\Delta}}({\mathfrak{Q}}_{k})$ -Baire
(${\tilde{\Delta}}({\mathfrak{Q}}_{k})$-nonmeager) space. For

$\tilde{{\tilde{\Delta}}}\in\\{\Delta_{k},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{e}$\hss}\,\,}}}{{{}_{k}}}{\operatorname{\Delta}},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{v}$\hss}\,\,}}}{{{}_{k}}}{\operatorname{\Delta}},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{c}$\hss}\,\,}}}{{{}_{k}}}{\operatorname{\Delta}},\Delta^{+}_{k},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{c}$\hss}\,\,}}}{{{}^{{}_{+}}_{k}}}{\operatorname{\Delta}},\Delta^{{}_{\gamma}}_{k},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{c}$\hss}\,\,}}}{{{}^{{}_{\gamma}}_{k}}}{\operatorname{\Delta}}\\}$

we have defined $\tilde{{\tilde{\Delta}}}$-Baire
($\tilde{{\tilde{\Delta}}}$-nonmeager) spaces. Also, by ${}^{k}\Delta^{l}$
without a subscript we mean ${}^{k}\Delta^{l}_{d}$: $\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{k}$\hss}\,\,}}}{{{}^{l}}}{\operatorname{\Delta}}$-nonmeager
($\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{k}$\hss}\,\,}}}{{{}^{l}}}{\operatorname{\Delta}}$-Baire)
means $\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{k}$\hss}\,\,}}}{{{}^{l}_{d}}}{\operatorname{\Delta}}$-nonmeager
($\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{k}$\hss}\,\,}}}{{{}^{l}_{d}}}{\operatorname{\Delta}}$-Baire).
Below we give separate direct definitions for the most important classes of
spaces: a space $X$ is called

  * (1)

$\Delta$-nonmeager ($\Delta$-Baire) if $X$ is
$\Delta(\mathfrak{S}(X);{\mathfrak{Q}}_{d})$-nonmeager
($\Delta(\mathfrak{S}(X);{\mathfrak{Q}}_{d})$-Baire);

  * (2)

$\Delta_{h}$-nonmeager ($\Delta_{h}$-Baire) if $X$ is
$\Delta(\mathfrak{S}(X);{\mathfrak{Q}}_{h})$-nonmeager
($\Delta(\mathfrak{S}(X);{\mathfrak{Q}}_{h})$-Baire);

  * (3)

$\Delta_{s}$-nonmeager ($\Delta_{s}$-Baire) if $X$ is
$\Delta(\mathfrak{S}(X);{\mathfrak{Q}}_{s})$-nonmeager
($\Delta(\mathfrak{S}(X);{\mathfrak{Q}}_{s})$-Baire);

  * (4)

$\Delta^{\gamma}$-nonmeager ($\Delta^{\gamma}$-Baire) if $X$ is
$\Delta({\mathsf{P}}_{\gamma}(\mathfrak{S}(X));{\mathfrak{Q}}_{d})$-nonmeager
($\Delta({\mathsf{P}}_{\gamma}(\mathfrak{S}(X));{\mathfrak{Q}}_{d})$-Baire);

  * (5)

$\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{c}$\hss}\,\,}}}{{{}^{\gamma}}}{\operatorname{\Delta}}$-nonmeager
($\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{c}$\hss}\,\,}}}{{{}^{\gamma}}}{\operatorname{\Delta}}$-Baire)
if $X$ is
$\Delta({\mathsf{P}}^{c}_{\gamma}(\mathfrak{S}(X));{\mathfrak{Q}}_{d})$-nonmeager
($\Delta({\mathsf{P}}^{c}_{\gamma}(\mathfrak{S}(X));{\mathfrak{Q}}_{d})$-Baire);

Since ${\mathsf{P}}_{+}(\mathfrak{S}(X))={\mathsf{P}}_{2}(\mathfrak{S}(X))$,
the classes $\Delta^{+}({\mathfrak{Q}})$-Baire
($\Delta^{+}({\mathfrak{Q}})$-nonmeager) and
$\Delta^{2}({\mathfrak{Q}})$-Baire ($\Delta^{2}({\mathfrak{Q}})$-nonmeager)
spaces coincide.

######  Remark 2.

The spaces that are here called ‘‘$\Delta$-nonmeager’’ were called
‘‘$\Delta$-Baire’’ in papers [2, 11, 12]. I decided to rename the class of
$\Delta$-Baire spaces because, as Proposition 34 shows, the $\Delta$-Baire
property is a generalization of the Baire property and the property of being
$\Delta$-nonmeager is a generalization of the property of being nonmeager.

Propositions 18, 20, 31 and Statement 10 imply the following assertion.

######  Proposition 33.

Let $X$ be a space, ${\mathfrak{Q}}$ be an NSF, and $n\in\omega$. Consider the
assertions:

  * (*)

    1. (a)

A space $X$ is ${\tilde{\Delta}}({\mathfrak{Q}})$-nonmeager if and only if
some nonempty open $U\subset X$ is ${\tilde{\Delta}}({\mathfrak{Q}})$-Baire.

    2. (b)

If the space $X$ is homogeneous, then $X$ is
${\tilde{\Delta}}({\mathfrak{Q}})$-nonmeager if and only if $X$ is
${\tilde{\Delta}}({\mathfrak{Q}})$-Baire.

  * (1)

If ${\tilde{\Delta}}\in\\{\Delta,\Delta^{n}\\}$, then (*) holds.

  * (2)

If the space $X$ is quasi-regular and
${\tilde{\Delta}}\in\\{\Delta,\Delta^{n},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{v}$\hss}\,\,}}}{{}}{\operatorname{\Delta}},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{c}$\hss}\,\,}}}{{}}{\operatorname{\Delta}},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{c}$\hss}\,\,}}}{{{}^{+}}}{\operatorname{\Delta}},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{c}$\hss}\,\,}}}{{{}^{n}}}{\operatorname{\Delta}}\\}$,
then (*) holds.

######  Statement 11.

Let $(X,{\mathcal{T}})$ be a quasi-regular space, and let
${{\mathcal{T}}^{*}}={\mathcal{T}}\setminus\\{\varnothing\\}$. A space $X$ is
meager if and only if there exists a sequence
$\gamma_{n}\subset{{\mathcal{T}}^{*}}$ of families of open sets for which the
following conditions are satisfied:

  1. 1.

$\gamma_{0}=\\{X\\}$;

  2. 2.

the family $\gamma_{n}$ is disjoint, $\overline{\bigcup\gamma_{n}}=X$ and
$\gamma_{n+1}$ refines $\gamma_{n}$;

  3. 3.

$\overline{U}\cap\overline{V}=\varnothing$ for different $U,V\in\gamma_{n}$;

  4. 4.

if ${n\in\omega}$, $V\in\gamma_{n+1}$, $U\in\gamma_{n}$ and $V\cap
U\neq\varnothing$, then $\overline{V}\subset U$;

  5. 5.

the family $\gamma=\bigcup\limits_{n\in\omega}\gamma_{n}$ is pointwise finite.

###### Proof.

If $\gamma_{n}$ with the listed properties exist, then
$F_{n}=X\setminus\bigcup\gamma_{n}$ is nowhere dense,
$X=\bigcup_{n\in\omega}F_{n}$, and hence $X$ is a meager space.

Suppose that $X$ is a meager space. Then there exist nowhere dense closed sets
$F_{n}\subset X$, $n\in\omega$ such that $X=\bigcup_{n\in\omega}F_{n}$. We can
assume that $F_{0}=\varnothing$ and $F_{n}\subset F_{n+1}$ for $n\in\omega$.
We construct $\gamma_{n}$ by induction on $n$.

We set $\gamma_{0}=\\{X\\}$. Suppose that $\gamma_{n}$ has been built. Let
$U\in\gamma_{n}$. It follows from Proposition 17 that there exists a disjoint
family $\nu_{U}\subset{{\mathcal{T}}^{*}}$ such that

  1. 1.

$\bigcup\nu_{U}\subset U\setminus F_{n+1}\subset\overline{\bigcup\nu_{U}}$;

  2. 2.

$\overline{V}\subset U\setminus F_{n+1}$ for $V\in\nu_{U}$;

  3. 3.

$\overline{W}\cap\overline{V}=\varnothing$ for different $W,V\in\nu_{U}$.

Let $\gamma_{n+1}=\bigcup\\{\nu_{U}\,:\,U\in\gamma_{n}\\}$. ∎

######  Statement 12.

Let $(X,{\mathcal{T}})$ be a meager quasi-regular space, and let
${{\mathcal{T}}^{*}}={\mathcal{T}}\setminus\\{\varnothing\\}$. Then there are
$P_{n}\in\mathfrak{S}(X)$ for $n\in\omega$ such that

  1. (1)

$P_{n}$ is nowhere dense in $X\times X$;

  2. (2)

$P_{0}\circ P_{n}=P_{n}$;

  3. (3)

$P_{n}\circ P_{n}\subset P_{2n}$;

  4. (4)

$\overline{P_{n}}\subset P_{n+1}$;

  5. (5)

$\bigcup\limits_{n\in\omega}P_{n}=X\times X$.

###### Proof.

Take $\gamma_{n}$, $n\in\omega$, as in Statement 11. We set

$\displaystyle f(x)$
$\displaystyle=\max\\{m\in\omega\,:\,x\in\bigcup\gamma_{m}\\},$ $\displaystyle
f_{n}(x)$ $\displaystyle=\max\\{0,f(x)-n\\}$

for $x\in X$ and $n\in\omega$. We set

$\displaystyle P_{n}=\\{(x,y)\in X\times X:\ $ $\displaystyle\text{there
exists a }U\in\gamma_{l},\text{ where }l=f_{n}(x),$ $\displaystyle\text{such
that }x,y\in U\\}.$

Let us check (1). Let $U,V\in{{\mathcal{T}}^{*}}$. Let $y\in V$. Let
$m=f(y)+n+1$. Take $W\in\gamma_{m}$ such that $U^{\prime}=W\cap
U\neq\varnothing$. Take $U^{\prime\prime}\in\gamma_{m-n}$ such that $W\subset
U^{\prime\prime}$. Then $P_{n}(U^{\prime})\subset U^{\prime\prime}$ and
$y\notin\overline{U^{\prime\prime}}$. Let
$V^{\prime}=V\setminus\overline{U^{\prime\prime}}$. Then $\varnothing\neq
U^{\prime}\times V^{\prime}\subset U\times V$ and $P_{n}\cap U^{\prime}\times
V^{\prime}=\varnothing$.

Let us check (2). Let $x\in X$, and let $m=f(x)$. If $m\leq n$, then
$P_{n}(x)=X$. Consider the case $m>n$. Take $V\in\gamma_{m}$ and
$U\in\gamma_{m-n}$ such that $x\in V\subset U$. Then $P_{0}(x)=V$,
$P_{n}(x)=U$ and $P_{n}(V)=U$.

Let us check (3). Let $x\in X$, and let $m=f(x)$. If $m\leq 2n$, then
$P_{2n}(x)=X\supset P_{n}(x)$. Consider the case $m>2n$. Take
$V\in\gamma_{m-n}$ and $U\in\gamma_{m-2n}$ such that $x\in V\subset U$. Then
$P_{n}(x)=V$, $P_{2n}(x)=U$ and $P_{n}(V)\subset U$.

Let us check (4). By construction, $\overline{P_{n}}^{v}\subset P_{n+1}$.
Proposition 9 implies that $\overline{P_{n}}\subset\overline{P_{0}\circ
P_{n}}^{v}$. Since, by virtue of (2), $P_{n}=P_{0}\circ P_{n}$, it follows
that $\overline{P_{n}}\subset P_{n+1}$.

Let us check (5). Let $x,y\in X$, $m=f(x)$. Then $P_{2m}(x)=X$ and $(x,y)\in
P_{2m}$. ∎

######  Proposition 34.

Let $X$ be a quasi-regular space, ${\mathfrak{Q}}$ be an NSF and $n\in\omega$.
If ${\tilde{\Delta}}\in\\{\Delta,\Delta^{n},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{v}$\hss}\,\,}}}{{}}{\operatorname{\Delta}},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{c}$\hss}\,\,}}}{{}}{\operatorname{\Delta}},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{c}$\hss}\,\,}}}{{{}^{+}}}{\operatorname{\Delta}},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{c}$\hss}\,\,}}}{{{}^{n}}}{\operatorname{\Delta}}\\}$,
then

  1. (1)

if $X$ is ${\tilde{\Delta}}({\mathfrak{Q}})$-nonmeager, then $X$ is nonmeager;

  2. (2)

if $X$ is ${\tilde{\Delta}}({\mathfrak{Q}})$-Baire, then $X$ is Baire.

###### Proof.

By Proposition 33 it suffices to prove (1). Assume that, on the contrary, $X$
is a meager space. Take $P_{n}\in\mathfrak{S}(X)$, $n\in\omega$, as in
Statement 12. If ${\mathcal{P}}=\mathfrak{S}(X)$ and

$\mathcal{Q}\in\\{{\mathcal{P}},{\mathsf{P}}^{v}({\mathcal{P}}),{\mathsf{P}}^{c}({\mathcal{P}}),{\mathsf{P}}_{+}^{c}({\mathcal{P}}),{\mathsf{P}}_{n}({\mathcal{P}}),{\mathsf{P}}_{n}^{c}({\mathcal{P}})\\},$

then $P_{m}\in{\mathsf{P}}^{e}(\mathcal{Q})$ for some $m\in\omega$. Since
$P_{m}$ is nowhere dense in $X\times X$, $X$ is not
$\Delta(\mathcal{Q};{\mathfrak{Q}})$-nonmeager. This contradiction proves the
prosition. ∎

######  Proposition 35.

Let $X$ be a space, and let $Y\subset X=\overline{Y}$.

  * (1)

Let ${\tilde{\Delta}}\in\\{\Delta,\Delta^{n},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{v}$\hss}\,\,}}}{{}}{\operatorname{\Delta}},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{c}$\hss}\,\,}}}{{}}{\operatorname{\Delta}},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{c}$\hss}\,\,}}}{{{}^{+}}}{\operatorname{\Delta}},\sideset{{}^{{\hbox
to0.0pt{$\scriptstyle{c}$\hss}\,\,}}}{{{}^{n}}}{\operatorname{\Delta}}\\}$. If
$Y$ is ${\tilde{\Delta}}$-nonmeager, then $X$ is ${\tilde{\Delta}}$-nonmeager.

  * (2)

Suppose that each point of the set $Y$ is a point of semi-regularity of the
space $X$ (for example, $X$ is a semi-regular space). If $Y$ is
$\Delta_{s}$-nonmeager, then $X$ is $\Delta_{s}$-nonmeager.

###### Proof.

Assertion (1) follows from Propositions 13 and 32. Let us prove (2). Let
$P\in\mathfrak{S}(X)$. Take $Q\in\mathfrak{S}(X)$ such that ${Q(y)}\subset
P(y)$ and $Q(y)$ is a regular open set for $y\in Y$. Since $Y$ is a
$\Delta_{s}$-nonmeager space, there exists a nonempty open $U\subset X$ such
that $R\in{\mathfrak{Q}}_{s}(S)$, where $S=Y\cap U$ and
$R=\left.Q\right|_{S}$.

######  Lemma.

Let $y\in S$. If $S=R(y)$, then $U\subset Q(y)\subset P(y)$.

###### Proof.

$U\subset\operatorname{Int}{\overline{S}}\subset\operatorname{Int}{\overline{R(y)}}\subset\operatorname{Int}{\overline{Q(y)}}=Q(y)$.
∎

The lemma, the relation $R\in{\mathfrak{Q}}_{s}(S)$, and Proposition 21 imply
$\left.P\right|_{U}\in{\mathfrak{Q}}_{s}(U)$. ∎

##  11 Relationship between $\Delta$ and $\Gamma$ Baire

In this section, a connection is established between Baire- and nonmeager-type
properties, determined by using the diagonal and topological games.

######  Proposition 36.

Let $(X,{\mathcal{T}})$ be a space, and let
${{\mathcal{T}}^{*}}={\mathcal{T}}\setminus\\{\varnothing\\}$.

  * (1)

If $X$ is $\Gamma^{\widehat{\scriptscriptstyle{OD}}}_{o,l}$-nonmeager
($\Gamma^{\widehat{\scriptscriptstyle{OD}}}_{o,l}$-Baire), then $X$ is
$\Delta$-nonmeager ($\Delta$-Baire).

  * (2)

If $X$ is $\Gamma^{\widehat{\scriptscriptstyle{OD}}}_{p,l}$-nonmeager
($\Gamma^{\widehat{\scriptscriptstyle{OD}}}_{p,l}$-Baire), then $X$ is
$\Delta_{h}$-nonmeager ($\Delta_{h}$-Baire).

  * (3)

If $X$ is $\Gamma^{\widehat{\scriptscriptstyle{BM}}}_{f}$-nonmeager
($\Gamma^{\widehat{\scriptscriptstyle{BM}}}_{f}$-Baire), then $X$ is
$\Delta_{s}$-nonmeager ($\Delta_{s}$-Baire).

###### Proof.

We will prove the proposition by contradiction based on Proposition 8. We
carry out the proof for generalizations of nonmeager spaces; for
generalizations of the Baire property, the proof is the same with the only
difference that we must additionally ensure during the construction that
$V_{0}=\varphi(X)$ is not generalized nonmeager.

(1) Assume that $X$ is not $\Delta$-nonmeager. Then there is an almost
neighborhood of the diagonal $P\in\mathfrak{S}(X)$ such that
$\left.P\right|_{U}\notin{\mathfrak{Q}}_{d}(U)$ for any
$U\in{{\mathcal{T}}^{*}}$. We define $(\varphi,\psi)\in
T_{OD}(X,{\mathcal{N}}_{o}(X))$. Recall that
${\mathcal{N}}_{o}(X)={{\mathcal{T}}^{*}}$. Let $U\in{{\mathcal{T}}^{*}}$.
Since $\left.P\right|_{U}\notin{\mathfrak{Q}}_{d}(U)$, it follows by
Proposition 21 that there exists $V\subset U$, $V\in{{\mathcal{T}}^{*}}$, such
that $U\not\subset\overline{P(V)}$. Let $\varphi(U)=V$ and $\psi(U)=M$, where
$M=U\setminus\overline{P(V)}$. Note that $M\cap P(V)=\varnothing$. We have
constructed mappings $\varphi$ and $\psi$. By Proposition 8, there exists
$(U_{n})_{n\in\omega}\in{\mathfrak{V}}(X)$ such that

  * (a)

$V_{n}=\varphi(U_{n})$, $M_{n}=\psi(U_{n})$, and $U_{n+1}\subset V_{n}$ for
${n\in\omega}$;

  * (b)

$(V_{n},M_{n})_{n\in\omega}\in{\mathfrak{W}}_{l}(X)$, i.e.,
$\operatorname{\overline{lt}}_{{n\in\omega}}M_{n}\cap\bigcap_{{n\in\omega}}V_{n}\neq\varnothing$.

The construction of $\psi$ implies

  * (c)

$M_{n}\cap P(V_{n})=\varnothing$ for any ${n\in\omega}$.

Let
$x_{*}\in\operatorname{\overline{lt}}_{{n\in\omega}}M_{n}\cap\bigcap_{{n\in\omega}}V_{n}$.
Then $M_{n}\cap P(x_{*})\neq\varnothing$ for some ${n\in\omega}$. We obtain
$M_{n}\cap P(V_{n})\neq\varnothing$. This contradictions (c).

(2) Assume that $X$ is not $\Delta_{h}$-nonmeager. Then there is an almost
neighborhood of the diagonal $P\in\mathfrak{S}(X)$ such that
$\left.P\right|_{U}\notin{\mathfrak{Q}}_{h}(U)$ for any
$U\in{{\mathcal{T}}^{*}}$. We define $(\varphi,\psi)\in
T_{OD}(X,{\mathcal{N}}_{p}(X))$. Recall that
${\mathcal{N}}_{p}(X)=\\{\\{x\\}\,:\,x\in X\\}$. Let
$U\in{{\mathcal{T}}^{*}}$. Since
$\left.P\right|_{U}\notin{\mathfrak{Q}}_{h}(U)$, it follows by Proposition 21,
there exists $V\subset U$, $V\in{{\mathcal{T}}^{*}}$, such that
$U\not\subset{P(V)}$. Let $\varphi(U)=V$ and $\psi(U)=\\{x\\}$, where $x\in
U\setminus{P(V)}$. Note that $x\notin P(V)$. We have constructed mappings
$\varphi$ and $\psi$. By Proposition 8, there exists
$(U_{n})_{n\in\omega}\in{\mathfrak{V}}(X)$ such that

  * (a)

$V_{n}=\varphi(U_{n})$, $\\{x_{n}\\}=\psi(U_{n})$, and $U_{n+1}\subset V_{n}$
for ${n\in\omega}$;

  * (b)

$(V_{n},\\{x_{n}\\})_{n\in\omega}\in{\mathfrak{W}}_{l}(X)$, i.e.,
$\operatorname{\overline{lt}}_{{n\in\omega}}x_{n}\cap\bigcap_{{n\in\omega}}V_{n}\neq\varnothing$.

The construction of $\psi$ implies

  * (c)

$x_{n}\notin P(V_{n})$ for any ${n\in\omega}$.

Let
$x_{*}\in\operatorname{\overline{lt}}_{{n\in\omega}}x_{n}\cap\bigcap_{{n\in\omega}}V_{n}$.
Then $x_{n}\in P(x_{*})$ for some ${n\in\omega}$. We obtain $x_{n}\in
P(V_{n})$. This contradicts (c).

(3) Assume that $X$ is not $\Delta_{s}$-nonmeager. Then there is an almost
neighborhood of the diagonal $P\in\mathfrak{S}(X)$ such that
$\left.P\right|_{U}\notin{\mathfrak{Q}}_{h}(U)$ for any
$U\in{{\mathcal{T}}^{*}}$. Let us define $\varphi\in T_{BM}(X)$. Let
$U\in{{\mathcal{T}}^{*}}$. Since
$\left.P\right|_{U}\notin{\mathfrak{Q}}_{s}(U)$, by Proposition 21, there
exists $V\subset U$, $V\in{{\mathcal{T}}^{*}}$, such that $U\not\subset P(x)$
for any $x\in V$. Let $\varphi(U)=V$. We have constructed a mapping
$\varphi:{{\mathcal{T}}^{*}}\to{{\mathcal{T}}^{*}}$. By Proposition 8, there
exists $(U_{n})_{n\in\omega}\in{\mathfrak{V}}(X)$ such that

  * (a)

$V_{n}=\varphi(U_{n})$, $U_{n+1}\subset V_{n}$ for ${n\in\omega}$;

  * (b)

$(V_{n})_{n\in\omega}\in{\mathfrak{V}}_{f}(X)$, i.e.,
$\bigcap_{{n\in\omega}}V_{n}\neq\varnothing$ and the family
$(V_{n})_{n\in\omega}$ is a base of some point
$x_{*}\in\bigcap_{{n\in\omega}}V_{n}$.

The construction of $\varphi$ implies

  * (c)

$U_{n}\not\subset P(x)$ for any ${n\in\omega}$ and $x\in V_{n}$.

Note that the family $(U_{n})_{n\in\omega}$ is also a base of the point
$x_{*}$. There exists ${n\in\omega}$ such that $U_{n}\subset P(x_{*})$. Since
$x_{*}\in V_{n}$, we obtain a contradiction to (c). ∎

Propositions 36, 30, 7, 4 and Theorems 2 and 3 imply the following theorem.

######  Theorem 4.

Let $X$ be a space. In the diagram below, the arrow

$A\to B$

means that

  * (1)

If $X$ is an $A$-nonmeager space, then $X$ is a $B$-nonmeager space;

  * (2)

if $X$ is an $A$-Baire space, then $X$ is a $B$-Baire space.

${\Gamma^{\scriptscriptstyle{BM}}_{f}}$${\Gamma^{\widehat{\scriptscriptstyle{BM}}}_{f}}$${\Delta_{s}}$${\Gamma^{\scriptscriptstyle{OD}}_{p,l}}$${\Gamma^{\widehat{\scriptscriptstyle{OD}}}_{p,l}}$${\Delta_{h}}$${\Gamma^{\scriptscriptstyle{OD}}_{o,l}}$${\Gamma^{\widehat{\scriptscriptstyle{OD}}}_{o,l}}$${\Delta}$

Let ${\mathcal{P}}_{k}$ (${\mathcal{P}}_{c}$) be the smallest class of spaces
that

  * 1.

contains $p$-spaces and strongly $\Sigma$-spaces;

  * 2.

is closed under arbitrary (countable) products;

  * 3.

is closed under taking open subspaces.

Given $\widetilde{\Delta}\in\\{\Delta_{s},\Delta_{h},\Delta\\}$, if a regular
Baire (nonmeager) space $X$ belongs to the class of spaces described in
$(\widetilde{\Delta})$, then $X$ is $\widetilde{\Delta}$-Baire
($\widetilde{\Delta}$-nonmeager).

  * $(\Delta_{s})$

$\sigma$-spaces.

  * $(\Delta_{h})$

is the smallest class of spaces that

    * (a)

contains $\Sigma$-spaces and $w\Delta$-spaces;

    * (b)

is closed under products by spaces from the class ${\mathcal{P}}_{c}$;

    * (c)

is closed under taking open subspaces.

  * $(\Delta)$

is the smallest class of spaces that

  * 1.

contains $\Sigma$-spaces, $w\Delta$-spaces, and feebly compact spaces;

  * 2.

is closed under products by spaces from the class ${\mathcal{P}}_{k}$;

  * 3.

is closed under taking open subspaces.

##  12 CDP-Baire spaces

Let $X$ be a space, and let $\mathcal{G}$ be a сollection of families of open
subsets of $X$. We denote

$\displaystyle B(X,\mathcal{G})\coloneqq\\{x\in X:\ $ the family
$\displaystyle\\{\operatorname{st}(x,\gamma)\,:\,\gamma\in\mathcal{G}\text{
and }x\in\bigcup\gamma\\}$ $\displaystyle\ \text{ is a base at }x\\}$
$\displaystyle\operatorname{dev}(X)\coloneqq\min\\{\,|\mathcal{G}|:\ $
$\displaystyle\mathcal{G}\text{ is a сollection of open covers of }X$
$\displaystyle\text{ and }X=B(X,\mathcal{G}),\\}.$

Let $Y\subset X$, and let $\mathcal{N}$ be the family of all nowhere dense
subsets of $X$. If $Y\not\subset\bigcup\mathcal{N}$, then we put
$\operatorname{Nov}(Y,X)\coloneqq\infty$, otherwise we put

$\displaystyle\operatorname{Nov}(Y,X)$
$\displaystyle\coloneqq\min\\{|\mathcal{L}|\,:\,\mathcal{L}\subset\mathcal{N}\text{
and }Y\subset\bigcup\mathcal{L}\\},$ $\displaystyle\operatorname{Nov}(X)$
$\displaystyle\coloneqq\operatorname{Nov}(X,X).$

######  Definition 13.

Let $X$ be a space.

  * 1.

We call a space a CDP-space if $\operatorname{dev}(X)<\operatorname{Nov}(X)$,
that is, if there exists a сollection ${\mathcal{P}}$ of open coverings of the
space $X$ such that $|{\mathcal{P}}|<\operatorname{Nov}(X)$ and
$B(X,{\mathcal{P}})=X$.

  * 2.

We call a space a CDP0-space if there exists a сollection ${\mathcal{P}}$ of
open partitions of $X$ such that $|{\mathcal{P}}|<\operatorname{Nov}(X)$ and
$B(X,{\mathcal{P}})=X$ .

  * 3.

We call a space $X$ CDP-nonmeager if there exists a сollection ${\mathcal{P}}$
of open families in $X$ such that $|{\mathcal{P}}|<\operatorname{Nov}(Y,X)$
for $Y=B(X,{\mathcal{P}})$.

  * 4.

We call a space $X$ CDP-Baire if every nonempty open subset of $X$ is a CDP-
nonmeager space.

The authors of [13] studied spaces for which
$\operatorname{dev}(X)<\operatorname{Nov}(X)$, that is, CDP-spaces. Metrizable
nonmeager spaces are CDP-spaces. Examples of non-metrizable CDP-spaces can be
obtained using Martin’s axiom ($MA$).

######  Proposition 37 (Theorem 2.3 [14]).

($MA$) Let $\tau<2^{\omega}$ be an infinite cardinal, and let $X$ be an
absolute $G_{\tau}$ space with countable Souslin number. Then
$\operatorname{Nov}(X)\geq 2^{\omega}$.

Proposition 37 implies the following assertion.

######  Proposition 38.

($MA$) Let $\tau<2^{\omega}$ be an infinite cardinal, and let $X$ be an
absolute $G_{\tau}$ space with countable Souslin number and
$\operatorname{dev}(X)\leq\tau$, for example $X=\mathbb{R}^{\tau}$. Then $X$
is a CDP-space.

Clearly, a CDP-space is a CDP0-space.

######  Proposition 39.

Let $X$ be a CDP-nonmeager space.

  * (1)

Then some nonempty open subset $U\subset X$ contains a dense CDP0-space
$Y\subset U\subset\overline{Y}$.

  * (2)

If $X$ is a $\pi$-semiregular space, then $U$ and $Y$ can be chosen so that
$X$ is semiregular at every point in $Y$.

###### Proof.

Let $\mathcal{G}=\\{\gamma_{\alpha}\,:\,\alpha<\tau\\}$ be a family of open
families in $X$ such that $\tau=|\mathcal{G}|<\operatorname{Nov}(S,X)$ for
$S=B(X,\mathcal{G})$.

Let $\alpha<\tau$. There is an open disjoint family $\nu_{\alpha}$ such that

  * 1.

if $U\in\nu_{\alpha}$ and $U\cap\bigcup\gamma_{\alpha}\neq\varnothing$, then
$U\subset V$ for some $V\in\gamma_{\alpha}$;

  * 2.

$\overline{\bigcup\nu_{\alpha}}=X$.

If $X$ is a $\pi$-semiregular space, then one can additionally ensure the
condition

  * 1.

the family $\nu_{\alpha}$ consists of regular open sets.

Let $G_{\alpha}=\bigcup\nu_{\alpha}$ and $F_{\alpha}=X\setminus G_{\alpha}$.
The set $F_{\alpha}$ is nowhere dense in $X$.

Let ${\mathcal{V}}=\\{\nu_{\alpha}\,:\,\alpha<\tau\\}$,
$G=\bigcap\\{G_{\alpha}\,:\,\alpha<\tau\\}$,
$F=\bigcup\\{F_{\alpha}\,:\,\alpha<\tau\\}$,

$Z=G\cap B(X,{\mathcal{V}}).$

Then $S\setminus F\subset Z$ and
$\operatorname{Nov}(Z,X)\geq\operatorname{Nov}(Y,X)>\tau$. Let
$U=\operatorname{Int}{\overline{Z}}$. Since $\operatorname{Nov}(Z,X)$ is
uncountable, $U\neq\varnothing$. Let $Y=Z\cap U$. The space $Y$ is a
CDP0-space, $Y\subset U\subset\overline{Y}$. If the families $\nu_{\alpha}$
consisted of regular open sets, then $X$ is semiregular at every point in $Y$.
∎

######  Proposition 40.

Let $X$ be a space, $U$ be an open subset of $X$, and $Y\subset
U\subset\overline{Y}\subset X$. If $Y$ is CDP-Baire and $X$ is semiregular at
points in $Y$, then $X$ is CDP-nonmeager.

###### Proof.

Take a сollection ${\mathcal{P}}$ of open in $Y$ coverings of the space $Y$
such that $|{\mathcal{P}}|<\operatorname{Nov}(Y)$ and $B(Y,{\mathcal{P}})=Y$.
For $\gamma\in{\mathcal{P}}$ we set
$\tilde{\gamma}=\\{\operatorname{Int}{\overline{V}}\,:\,V\in\gamma\\}$. We put
$\widetilde{{\mathcal{P}}}=\\{\tilde{\gamma}\,:\,\gamma\in{\mathcal{P}}\\}$.
Then $Y\subset\widetilde{Y}=B(X,\widetilde{{\mathcal{P}}})$ and
$|\widetilde{{\mathcal{P}}}|<\operatorname{Nov}(Y)\leq\operatorname{Nov}(\widetilde{Y},X)$.
∎

Propositions 39 and 40 imply the following theorem.

######  Theorem 5.

Let $X$ be a semiregular space. The following conditions are equivalent:

  * (1)

$X$ is a CDP-Baire space;

  * (2)

some nonempty open subset $U\subset X$ contains a dense CDP-space $Y\subset
U\subset\overline{Y}$;

  * (3)

some nonempty open subset $U\subset X$ contains a dense CDP0-space $Y\subset
U\subset\overline{Y}$.

Theorem 5 implies

######  Proposition 41.

Let $X$ be a semiregular space. If $X$ has a metrizable nonmeager subspace,
then $X$ is a CDP-nonmeager space.

##  13 $\Delta_{s}$-Baire spaces

Theorem 4 implies that a $\Gamma^{\scriptscriptstyle{BM}}_{f}$-Baire
($\Gamma^{\scriptscriptstyle{BM}}_{f}$-nonmeager) space is a
$\Delta_{s}$-Baire ($\Delta_{s}$-nonmeager) space. Proposition 4 and theorem 2
give the following examples of $\Gamma^{\scriptscriptstyle{BM}}_{f}$-Baire
($\Gamma^{\scriptscriptstyle{BM}}_{f}$-nonmeager) spaces; Baire (nonmeager)
spaces from the classes of spaces listed below are
$\Gamma^{\scriptscriptstyle{BM}}_{f}$-Baire
($\Gamma^{\scriptscriptstyle{BM}}_{f}$-nonmeager):

  * metrizable spaces, Moore spaces, developable space, semiregular $\sigma$-spaces, and semiregular spaces with a countable network.

######  Proposition 42.

Let $X$ be a space. If $X$ is CDP-nonmeager (CDP-Baire), then $X$ is
$\Delta_{s}$-nonmeager ($\Delta_{s}$-Baire).

###### Proof.

Let us prove the propositions for the case when $X$ is CDP-nonmeager. Let
$\mathcal{G}$ be a family of open families in $X$ such that
$|\mathcal{G}|<\operatorname{Nov}(Y,X)$ for $Y=B(X,\mathcal{G})$. Let
$P\in\mathfrak{S}(X)$ be an almost neighborhood of the diagonal $X\times X$.
For every $y\in Y$ there is $\gamma_{y}\in\mathcal{G}$ such that
$\operatorname{st}(y,\gamma_{y})\subset P(y)$. We put $Y_{\gamma}=\\{y\in
Y\,:\,\gamma_{y}=\gamma\\}$. Since $|\mathcal{G}|<\operatorname{Nov}(Y,X)$ ,
it follows that $Y_{\gamma}^{\prime}$ is nowhere dense for some
$\gamma^{\prime}\in\mathcal{G}$. We choose $y^{\prime}\in V\cap Y$, where
$V=\operatorname{Int}(\overline{Y_{\gamma^{\prime}}})$. Let
$W\in\operatorname{St}(y^{\prime},\gamma^{\prime})$. Let $U=W\cap V$ and
$M=U\cap Y$. Then $M$ is dense in $U$ and for $y\in M$

$U\subset W\subset\operatorname{st}(y,\gamma^{\prime})\subset P(y).$

Hence $M\times U\subset P$. ∎

Examples of $\Delta_{s}$-nonmeager spaces that are obtained from
$\Gamma^{\scriptscriptstyle{BM}}_{f}$-nonmeager spaces always contain points
with the first countability axiom. Propositions 42 and 38 imply that under the
assumption MA+$\lnot$CH the space $\mathbb{R}^{\omega_{1}}$ is
$\Delta_{s}$-nonmeager.

##  14 Continuity of operations in groups

The main applications of the considered generalizations of Baire spaces relate
to the topic of continuity of operations in groups with topology. Such
applications were considered in [3]. In this section, we formulate results
that will help to distinguish between different types of Baire.

Let $G$ be a topological group. A group $G$ is called semitopological if
multiplication in $G$ is separately continuous, that is, for $g\in G$ the maps
$x\mapsto gx$ and $x\mapsto xg$ are continuous. A group $G$ is called
paratopological if the multiplication in $G$ is continuous. A group $G$ is
called quasitopological if $G$ is a semitopological group and the operation of
taking the inverse of $x\mapsto x^{-1}$ is continuous. A group $G$ is called
topological if the operations in $G$ are continuous, that is, $G$ is a
paratopological quasi-topological group.

######  Proposition 43 ([2]).

Let $G$ be a regular $\Delta$-nonmeager paratopological group. Then $G$ is a
topological group.

######  Proposition 44.

Let $G$ be a semiregular $\Delta_{h}$-nonmeager quasitopological group. Then
$G$ is a topological group.

###### Proof.

Let $e$ be the identity element of the group $G$, and let $O\ni e$ be an open
neighborhood of the identity element. We need to find a neighborhood $V$ for
which $VV\subset O$.

Let $U\ni e$ be a regular open symmetric neighborhood of the identity element.
We put $P=\\{(g,h)\in G\times G\,:\,h\in gU\\}$. Then $P$ is a semi-
neighborhood of the diagonal and $P(g)=gU$ for $g\in G$. Since $G$ is
$\Delta_{h}$-nonmeager and $P=P^{-1}$, there is an open nonempty $W\subset G$
such that $W\subset\overline{P(x)}$ for $x\in W$. Hence
$WW^{-1}\subset\overline{U}$. Since $U=\operatorname{Int}\overline{U}$ and the
set $WW^{-1}$ is open, $WW^{-1}\subset U$. Let $V\ni e$ be a symmetric
neighborhood of the identity element such that $Vg\subset W$ for some $g\in
W$. Then $VV\subset U\subset O$. ∎

######  Proposition 45 (Theorem 4 [15]).

Let $X$ be a space such that $X^{\omega}$ is a pseudocompact space. Let $H$ be
an Abelian group such that $|X|\leq|H|=|H|^{\omega}$. Then there exists a
semitopological group $G$ such that $G^{\omega}$ is a pseudocompact space, $G$
maps continuously onto $X$, and there exists a homomorphism $H\to G$.

Recall that a group is called Boolean if it satisfies the identity $g^{2}=e$.
In a Boolean group, taking the inverse element is the identity operation. Let
$X$ be a compact space with uncountable Souslin number and $H$ a Boolean group
such that $|X|\leq|H|=|H|^{\omega}$. Then from Proposition 45 we obtain the
following corollary.

######  Corollary 1 (Korovin).

There exist a pseudocompact Boolean quasitolpological group which is not a
topological group.

##  15 Examples and questions

In this section, we study how different the introduced classes of spaces are.
This section suplements the corresponding section in [1].

The following diagram shows the relationship between the most interesting
classes of spaces.

Any arrow $A\to B$ means that any $A$-Baire space is a $B$-Baire space and
${A}$${B}$ and that the converse is not true; arrow ${A}$${B}$ means that it
is not known whether or not the converse is true; arrow ${A}$${B}$ means that
there is a counterexample for the converse assertion under additional set-
theoretic asseumptions.

The following diagram summarizes Propositions 3 and 42 and Theorem 4.
Counterexamples are given in [1] and will be constructed below for the bottom
line of the diagram.

${\Gamma^{\scriptscriptstyle{BM}}_{o}}$${\Gamma^{\scriptscriptstyle{BM}}_{p}}$${\Gamma^{\scriptscriptstyle{BM}}_{k}}$${\Gamma^{\scriptscriptstyle{OD}}_{o,k}}$${\Gamma^{\scriptscriptstyle{OD}}_{p,k}}$${\Gamma^{\scriptscriptstyle{OD}}_{o,l}}$${\Gamma^{\scriptscriptstyle{OD}}_{p,l}}$${\Gamma^{\scriptscriptstyle{BM}}_{f}}$${\Delta}$${\Delta_{h}}$${\Delta_{s}}$${\mathrm{CDP}}$

We denote by $\mathbb{D}$ the discrete two-point space $\\{0,1\\}$. The base
of the topology in $\mathbb{D}^{C}$ is formed by sets of the form

$W(A,B,C)\coloneqq\\{(x_{\alpha})_{\alpha\in
C}\in\mathbb{D}^{C}\,:\,x_{\alpha}=0\text{ for }\alpha\in A\text{ and
}x_{\beta}=1\text{ for }\beta\in B\\}$

for finite disjoint $A,B\subset C$.

Propositions 4, 1 and Theorems 2 and 3 imply the following assertion.

######  Statement 13 (Assertion 11 [1]).

Let $X$ be a regular space without isolated points.

  * (1)

If $X$ is compact, then $X$ is $\Gamma^{\scriptscriptstyle{BM}}_{k}$-Baire.

  * (2)

If $X$ is $\Gamma^{\scriptscriptstyle{OD}}_{o,k}$-nonmeager, then $X$ contains
an infinite compact set.

  * (3)

If $X$ is countably compact, then $X$ is
$\Gamma^{\scriptscriptstyle{BM}}_{p}$-Baire.

  * (4)

If $X$ is $\Gamma^{\scriptscriptstyle{OD}}_{p,l}$-nonmeager, then $X$ contains
a non-discrete countable space.

  * (5)

If $X$ is pseudocompact, then $X$ is
$\Gamma^{\scriptscriptstyle{BM}}_{o}$-Baire.

  * (6)

If $X$ is $\Gamma^{\scriptscriptstyle{BM}}_{f}$-nonmeager, then $X$ contains
points with a countable base.

  * (7)

If $X$ is $\Gamma^{\scriptscriptstyle{BM}}_{o}$-nonmeager, then $X$ contains
$q_{o}$-points.

  * (8)

If $X$ is a product of locally compact spaces (for example,
$X=\mathbb{R}^{\tau}$), then $X$ is
$\Gamma^{\scriptscriptstyle{OD}}_{p,k}$-Baire.

Below we give examples distinguishing between the classes of spaces listed
above. In the examples below, $A\nrightarrow B$: $X$ means that $X$ is an
$A$-Baire space that is not $B$-nonmeager.

######  Example 1.

$\mathrm{CDP}\nrightarrow\Gamma^{\scriptscriptstyle{BM}}_{f}$:
$\mathbb{R}^{\omega_{1}}$ (assuming MA+$\lnot$CH). Assertion 13 (7) and
Proposition 38.

######  Example 2.

$\Delta_{s}\nrightarrow\Gamma^{\scriptscriptstyle{BM}}_{k}$:
$\mathbb{D}^{2^{\omega}}$. We denote $X=\mathbb{D}^{2^{\omega}}$. The space
$X$ is $\Gamma^{\scriptscriptstyle{BM}}_{k}$-Baire (Statement 13 (1)). Let us
show that $X$ is not $\Delta_{s}$-nonmeager. There is a cover
$\\{F_{\alpha}\,:\,\alpha<2^{\omega}\\}$ of the space $X$ of size
$2^{\omega}$, consisting of nowhere dense closed sets. We fix
$\alpha_{x}<2^{\omega}$ for $x\in X$ such that $x\in F_{\alpha_{x}}$. We set
$\tilde{x}=(\tilde{x}_{\alpha})_{\alpha\in 2^{\omega}}$, where
$\tilde{x}_{\alpha}=x_{\alpha}$ for $\alpha\neq\alpha_{x}$ and
$\tilde{x}_{\alpha_{x}}\neq x_{\alpha_{x}}$. We define an almost neighborhood
$P$ of the diagonal $X\times X$, by setting $P(x)=X\setminus\\{\tilde{x}\\}$.
Let us show that $\left.P\right|_{V}\notin{\mathfrak{Q}}_{s}(V)$ for any open
nonempty $V\subset X$. Let $U=W(A,B,2^{\omega})\subset V$ for some finite
disjoint $A,B\subset 2^{\omega}$. We set

$W=V\setminus\bigcup_{\alpha\in A\cup B}F_{\alpha}.$

Then $W$ is dense and open in $U$ and $\tilde{x}\in U$ for $x\in W$.
Consequently

$M=\\{x\in U\,:\,U\subset P(x)\\}\cap W=\varnothing$

and the set $M$ is nowhere dense in $U$.

######  Example 3.

$\Gamma^{\scriptscriptstyle{BM}}_{o}\nrightarrow\Delta_{h}$: $G$. Let $G$ be a
pseudocompact Boolean quasitolpological group that is not a topological group
(Corollary 1). The space $G$ is a $\Gamma^{\scriptscriptstyle{BM}}_{o}$-Baire
space. Proposition 44 implies that $G$ is not a $\Delta_{h}$-nonmeager space.

A space $X$ is called weakly pseudocompact if there exists a compact Hausdorff
extension $bX$ of the space $X$ in which the space $X$ is $G_{\delta}$-dense,
i.e., $X$ intersects any nonempty $G_{\delta}$ subset of $bX$ [16]. It is
clear that a product of weakly pseudocompact spaces is weakly pseudocompact;
in particular, a product of pseudocompact spaces is weakly pseudocompact.

######  Problem 1.

Let $X$ be a weakly pseudocompact space (or a product of pseudocompact
spaces). To which of the following classes does $X$ belong:

  * $\Gamma^{\scriptscriptstyle{BM}}_{o}$-spaces, $\Gamma^{\scriptscriptstyle{BM}}_{o}$-Baire spaces, $\Gamma^{\scriptscriptstyle{OD}}_{o,l}$-spaces, $\Gamma^{\scriptscriptstyle{OD}}_{o,l}$-Baire spaces, $\Delta$-Baire spaces?

A paratological regular $\Delta$-nonmeager group is a topological group
(Proposition 43).

######  Problem 2.

Let $G$ be a paratopological group which is a weakly pseudocompact space (a
product of pseudocompact spaces). Is it true that $G$ is a topological group?

######  Problem 3.

Let $X$ and $Y$ be (completely) regular countably compact spaces. To which of
the following classes does the product $X\times Y$ belong:

  * $\Gamma^{\scriptscriptstyle{BM}}_{p}$-spaces, $\Gamma^{\scriptscriptstyle{BM}}_{p}$-Baire spaces, $\Gamma^{\scriptscriptstyle{OD}}_{p,l}$-spaces, $\Gamma^{\scriptscriptstyle{OD}}_{p,l}$-Baire spaces, $\Gamma^{\scriptscriptstyle{BM}}_{o}$-spaces, $\Gamma^{\scriptscriptstyle{BM}}_{o}$-Baire spaces, $\Gamma^{\scriptscriptstyle{OD}}_{o,l}$-spaces, $\Gamma^{\scriptscriptstyle{OD}}_{o,l}$-Baire spaces, $\Delta_{h}$-Baire spaces, $\Delta$-Baire spaces?

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