# Power homogeneous compacta and variations on tightness

Nathan Carlson  Department of Mathematics, California Lutheran University, 60
W. Olsen Rd, MC 3750, Thousand Oaks, CA 91360 USA<EMAIL_ADDRESS>
###### Abstract.

The weak tightness $wt(X)$, introduced in [6], has the property $wt(X)\leq
t(X)$. It was shown in [4] that if $X$ is a homogeneous compactum then
$|X|\leq 2^{wt(X)\pi\chi(X)}$. We introduce the almost tightness $at(X)$ with
the property $wt(X)\leq at(X)\leq t(X)$ and show that if $X$ is a power
homogeneous compactum then $|X|\leq 2^{at(X)\pi\chi(X)}$. This improves the
result of Arhangel′skiĭ, van Mill, and Ridderbos in [2] that $|X|\leq
2^{t(X)}$ for a power homogeneous compactum $X$ and gives a partial answer to
a question in [4]. In addition, if $X$ is a homogeneous Hausdorff space we
show that $|X|\leq 2^{pw_{c}L(X)wt(X)\pi\chi(X)pct(X)}$, improving a result in
[3]. It also extends the result in [4] into the Hausdorff setting. The
cardinal invariant $pwL_{c}(X)$, introduced in [5] by Bella and Spadaro,
satisfies $pwL_{c}(X)\leq L(X)$ and $pwL_{c}(X)\leq c(X)$. We also show the
weight $w(X)$ of a homogeneous space $X$ is bounded in various contexts using
$wt(X)$. One such result is that if $X$ is homogeneous and regular then
$w(X)\leq 2^{L(X)wt(X)pct(X)}$. This generalizes a result in [4] that if $X$
is a homogeneous compactum then $w(X)\leq 2^{wt(X)}$.

###### Key words and phrases:

cardinality bounds, cardinal invariants, homogeneous space, weak tightness

###### 2010 Mathematics Subject Classification:

54A25, 54B10

##  1\. Introduction.

Motivated by results of Juhász and van Mill in [13], the _weak tightness_
$wt(X)$ of a topological space $X$ was introduced in [6] with the property
$wt(X)\leq t(X)$ (see Definition LABEL:weakTightness). This cardinal invariant
was investigated further by Bella and the author in [4], where the following
theorem was shown. Recall a space $X$ is _homogeneous_ if for all $x,y\in X$
there exists a homeomorphism $h:X\to X$ such that $h(x)=y$. $X$ is _power
homogeneous_ if there exists a cardinal $\kappa$ such that $X^{\kappa}$ is
homogeneous.

######  Theorem 1.1 ([4]).

Let $X$ be a homogeneous compactum. Then,

  * (a)

$w(X)\leq 2^{wt(X)}$, and

  * (b)

$|X|\leq 2^{wt(X)\pi\chi(X)}$.

As $\pi\chi(X)\leq t(X)$ for a compactum $X$ (Šapirovskiĭ [18]), Theorem
LABEL:cpthomog(b) improved a result of de la Vega [10] that the cardinality of
a homogeneous compactum is at most $2^{t(X)}$.

It was asked in Question 3.13 in [4] if $2^{wt(X)\pi\chi(X)}$ is a bound for
the cardinality of a _power_ homogeneous compactum $X$. While this question is
still open, we attain a partial answer by introducing a cardinal invariant
related to $wt(X)$, the _almost tightness_ $at(X)$ of a space $X$ (see
Definition LABEL:almostTightness). The almost tightness, satisfying $wt(X)\leq
at(X)\leq t(X)$, resolves technical issues when considering power homogeneous
spaces that are not resolved with $wt(X)$ (see section 2). We show in Theorem
LABEL:cptPH that $|X|\leq 2^{at(X)\pi\chi(X)}$ for a power homogeneous
compactum $X$. This improves a result of Arhangel′skiĭ, van Mill, and
Ridderbos [2] that $|X|\leq 2^{t(X)}$ for such spaces. Theorem LABEL:cptPH is
related to a result of Juhász and van Mill [13] and also to a result in [6].
See section 3 for the details of these relationships.

In section 4 we generalize Theorem LABEL:cpthomog into the Hausdorff and
regular settings. The first step in this extension process is Theorem
LABEL:compactsubset2, which establishes that if $X$ is Hausdorff and
$wt(X)pct(X)\leq\kappa$, then there exists a nonempty compact set $G\subseteq
X$ and $H\subseteq X$ such that $\chi(G,X)\leq\kappa$,
$G\subseteq\overline{H}$, $|H|\leq 2^{\kappa}$, and $H$ is
$\mathcal{C}$-saturated. (See LABEL:pct for the definition of $pct(X)$ and
LABEL:Csat for the definition of $\mathcal{C}$-saturated). This represents an
extension of Theorem 3.3 in [4] into the Hausdorff setting. That theorem
established the existence of $G$ and $H$ with the above properties when $X$ is
compact and $wt(X)\leq\kappa$.

We use the cardinal invariant $pwL_{c}(X)$, first defined in [5] (see
Definition LABEL:pwL below). This invariant has the properties $pwL_{c}(X)\leq
L(X)$ and $pwL_{c}(X)\leq c(X)$. It was shown in [5] that if $X$ is Hausdorff
then $|X|\leq 2^{pwL_{c}(X)\chi(X)}$, unifying the well-known bounds
$2^{L(X)\chi(X)}$ and $2^{c(X)\chi(X)}$ for the cardinality of a Hausdorff
space $X$. We show in Theorem LABEL:pbound that if $X$ is additionally
homogeneous then $|X|\leq 2^{pwL_{c}(X)wt(X)\pi\chi(X)pct(X)}$. This improves
Corollary 3.7 in [3], which states that $|X|\leq 2^{pwL_{c}(X)t(X)pct(X)}$ if
$X$ is homogeneous and Hausdorff. Theorem LABEL:pbound also generalizes
Theorem LABEL:cpthomog(b) into the class of Hausdorff spaces.

We also extend Theorem LABEL:cpthomog(a) into the regular setting by
demonstrating that if $X$ is a homogeneous, regular, Hausdorff space then
$w(X)\leq 2^{L(X)wt(X)pct(X)}$.

By _compactum_ we mean a compact, Hausdorff space. For all undefined notions
see the books [11] and [12] by Engelking and Juhász, respectively. We make no
implicit assumption of any separation axiom on a space $X$.

##  2\. Tools for homogeneous product spaces.

In this section we modify tools given in [2] and in [17] for product spaces
$X$ and homeomorphisms $h:X\to X$. Variations of these tools will be used to
establish a result for power homogeneous spaces, Corollary LABEL:GPH, that
will be needed in the proof of Theorem LABEL:cptPH.

For this section we fix the following notation. Let $X=\prod\\{X_{i}:i\in
I\\}$ be endowed with the product topology, where $X_{i}$ is a space for all
$i\in I$. Let $H=\prod\\{H_{i}:i\in I\\}$ where $H_{i}\subseteq X_{i}$, and
for all $i\in I$, fix $s_{i}\in H_{i}$. If $A\subseteq I$, let

$H(A)=\\{x\in H:x_{i}=s_{i}\textup{ except for finitely many }i\in A\\}.$

When $K\subseteq X$, let $K_{A}=\pi_{A}[K]$, where
$\pi_{A}:X\to\prod\\{X_{i}:i\in A\\}$ is the projection map. If $x\in X$ and
$\pi_{j}:X\to X_{j}$ is the projection for some $j\in I$, then we set
$x_{j}=\pi_{j}(x)$.

The following is Corollary 2.9 in [2], re-worded to fit our notation.

######  Theorem 2.1 (Corollary 2.9 [2]).

Suppose that $X$ is homogeneous and for all $i\in I$ there exists a non-empty
closed $G_{\kappa}$-set $G_{i}$ and a set $H_{i}\in[X_{i}]^{\leq\kappa}$ such
that $G\subseteq\overline{H_{i}}$. If for some $j\in I$ we have
$\pi\chi(X_{j})\leq\kappa$, then there exists a cover $\mathcal{G}$ of $X_{j}$
consisting of closed $G_{\kappa}$-sets such that for all $G\in\mathcal{G}$
there exists $H_{G}\in[X_{j}]^{\leq\kappa}$ such that
$G\subseteq\overline{H_{G}}$.

A set $G$ is a $G^{c}_{\kappa}$-_set_ of a space $Y$ if there exists a family
of open sets $\mathcal{U}$ in $Y$ such that $|\mathcal{U}|\leq\kappa$ and
$G=\bigcap\mathcal{U}=\bigcap_{U\in\mathcal{U}}\overline{U}$. Note that a
$G^{c}_{\kappa}$-set is a special type of closed $G_{\kappa}$-set. Observe
that a compact $G_{\kappa}$-set in a Hausdorff space $Y$ is a
$G^{c}_{\kappa}$-set. Our aim in this section is to ensure that if each
$G_{i}$ in the Theorem LABEL:AVR is a $G^{c}_{\kappa}$-set in $X_{i}$ then
each member of the cover $\mathcal{G}$ is a $G^{c}_{\kappa}$-set in $X_{j}$.
We accomplish this in Corollary LABEL:PH1.

The following are two results from [2]. The first gives straightforward
properties of the set $H(A)$ for $A\in[I]^{\leq\kappa}$. The second gives
conditions for the existence of a set $A\in[I]^{\leq\kappa}$ with certain
properties, given a homeomorphism $h:X\to X$.

######  Lemma 2.2 (Lemma 2.7 [2]).

If $A\subseteq I$ is such that $|A|\leq\kappa$ for some cardinal $\kappa$, and
for all $i\in A$, $|H_{i}|\leq\kappa$, then $|H(A)|\leq\kappa$ and $H(A)_{A}$
is dense in $H_{A}$. Futhermore, if $A\subseteq B$ then $H(A)\subseteq H(B)$.
If $(A_{n})$ is an increasing sequence of infinite subsets of $I$ and
$A=\bigcup_{n=1}^{\infty}A_{n}$, then $H(A)=\bigcup_{n=1}^{\infty}H(A_{n})$.

######  Theorem 2.3 (Theorem 2.8 [2]).

Suppose there exists a cardinal $\kappa$ such that $|H_{i}|\leq\kappa$ for all
$i\in I$. Suppose that for some $j\in I$, $\pi\chi(X_{j})\leq\kappa$. If
$h:X\to X$ is a homeomorphism and $B\subseteq I$ such that $|B|\leq\kappa$,
then there is a set $A\subseteq I$ such that $|A|\leq\kappa$, $B\subseteq A$,
and for all $s\in H(A)$ and for all $x\in X$, if $x_{A}=s_{A}$ then
$h(s)_{j}=h(x)_{j}$.

We obtain the following new result, which is an adjustment to Corollary 2.9 in
[2].

######  Corollary 2.4.

Suppose $X$ is homogeneous and Hausdorff and suppose for all $i\in I$ there
exist $e_{i}\in X_{i}$, a $G^{c}_{\kappa}$-set $G_{i}\subseteq X_{i}$, and
$H_{i}\subseteq X_{i}$ such that $|H_{i}|\leq\kappa$ and $e_{i}\subseteq
G_{i}\subseteq\overline{H_{i}}$. If for some $j\in I$ we have
$\pi\chi(X_{j})\leq\kappa$, then for all $z\in X_{j}$ there exists a
$G^{c}_{\kappa}$-set $F$ in $X_{j}$ and $T\subseteq X_{j}$ such that
$|T|\leq\kappa$ and $z\in F\subseteq\overline{T}$.

###### Proof.

Let $G=\prod\\{G_{i}:i\in I\\}$ and let $z\in X_{j}$ be arbitrary. Pick $w\in
X$ such that $w_{j}=z$. As $X$ is homogeneous, there exists a homeomorphism
$h:X\to X$ such that $h(e)=w$, where $e$ is the point of $X$ whose $i^{th}$
coordinate is $e_{i}$. By Theorem LABEL:getA2kappa and Lemma LABEL:S(A) we
obtain $A\subseteq I$ such that $|A|\leq\kappa$ and

  1. (1)

$|H(A)|\leq\kappa$,

  2. (2)

$e_{A}\in G_{A}\subseteq\overline{H(A)_{A}}$, and

  3. (3)

for all $s\in H(A)$ and all $x\in X$ if $x_{A}=s_{A}$ then
$h(x)_{j}=h(s)_{j}$.

Now, for all $s\in H(A)$ the set $\pi_{j}h\pi_{A}^{-1}(s_{A})$ consists of the
single point $h(s)_{j}$. Let $T=\\{h(s)_{j}:s\in H(A)\\}$ and note
$T\in[X_{j}]^{\leq\kappa}$. Since $G_{A}\subseteq\overline{H(A)_{A}}$ and
$\pi_{A}$ is open, we have
$\pi_{A}^{-1}[G_{A}]\subseteq\overline{\pi_{A}^{-1}[H(A)_{A}]}$. Thus

$\pi_{j}h[\pi_{A}^{-1}[G_{A}]]\subseteq\overline{\pi_{j}h\pi_{A}^{-1}[H(A)_{A}]}=\overline{T}.$

We show that $\pi_{A}[G_{A}]$ is a $G^{c}_{\kappa}$-set in $X$. As each
$G_{i}$ is a $G^{c}_{\kappa}$-set in $X_{i}$, for all $i\in I$ there exists a
family $\\{U(i,\alpha):\alpha<\kappa\\}$ of open sets in $X_{i}$ such that
$G_{i}=\bigcap_{\alpha<\kappa}U(i,\alpha)=\bigcap_{\alpha<\kappa}\overline{U(i,\alpha)}$.
Now,

(2.1) $\displaystyle\pi_{A}^{-1}[G_{A}]$ $\displaystyle=\prod_{i\in
A}G_{i}\times\prod_{i\notin A}X_{i}=\prod_{i\in
A}(\bigcap_{\alpha<\kappa}U(i,\alpha))\times\prod_{i\notin A}X_{i}$
$\displaystyle=\bigcap_{\alpha<\kappa}(\prod_{i\in
A}U(i,\alpha)\times\prod_{i\notin
A}X_{i})=\bigcap_{\alpha<\kappa}\bigcap_{i\in A}(U(i,\alpha)\times\prod_{l\neq
i}X_{l}).$

Therefore $\pi_{A}^{-1}[G_{A}]$ is the intersection of $\kappa$-many open sets
in $X$, as $|A|\leq\kappa$. Similarly, it can be shown that
$\pi_{A}^{-1}[G_{A}]=\bigcap_{\alpha<\kappa}\bigcap_{i\in
A}\overline{U(i,\alpha)\times\prod_{l\neq i}X_{l}}$, making
$\pi_{A}^{-1}[G_{A}]$ a $G^{c}_{\kappa}$-set in $X$. It follows that
$h[\pi_{A}^{-1}[G_{A}]]$ is also a $G^{c}_{\kappa}$-set in $X$ as $h$ is a
homeomorphism.

Let $B=I\backslash\\{j\\}$ and $Y=\\{y\in X:y_{B}=w_{B}\\}$. Then
$\left.\pi_{j}\right|_{Y}:Y\to X_{j}$ is a homeomorphism. Now let $K=Y\cap
h[\pi_{A}^{-1}[G_{A}]]$. We show $K$ is a $G^{c}_{\kappa}$-set of $Y$. There
exists a family $\mathcal{U}$ of open sets of $X$ such that
$|\mathcal{U}|\leq\kappa$ and
$h[\pi_{A}^{-1}[G_{A}]]=\bigcap\mathcal{U}=\bigcap_{U\in\mathcal{U}}\overline{U}$.
Then,

$K=Y\cap\bigcap\mathcal{U}=\bigcap_{U\in\mathcal{U}}(U\cap
Y)\subseteq\bigcap_{U\in\mathcal{U}}(\overline{U\cap Y}\cap
Y)\subseteq\bigcap_{U\in\mathcal{U}}(\overline{U}\cap Y)=K,$

and therefore $K=\bigcap_{U\in\mathcal{U}}(U\cap
Y)=\bigcap_{U\in\mathcal{U}}(\overline{U\cap Y}\cap
Y)=\bigcap_{U\in\mathcal{U}}cl_{Y}(U\cap Y)$.

This shows $K$ is a $G^{c}_{\kappa}$-set of $Y$. Let $F=\pi_{j}[K]$. Then $F$
is a $G^{c}_{\kappa}$-set of $X_{j}$ as $\left.\pi_{j}\right|_{Y}:Y\to X_{j}$
is a homeomorphism. We have,

$z=w_{j}\in
F=\pi_{j}[K]\subseteq\pi_{j}h[\pi_{A}^{-1}[G_{A}]]\subseteq\overline{T}.$

Thus $z\in F\subseteq\overline{T}$. This completes the proof. ∎

For our last corollary in this section we dispense with the convention that
$X$ is a product space $\prod\\{X_{i}:i\in I\\}$ and that
$H=\prod\\{H_{i}:i\in I\\}$. It follows immediately from Corollary LABEL:PH1.
Compare this to Theorem LABEL:AVR.

######  Corollary 2.5.

Let $X$ be a power homogeneous Hausdorff space. Suppose there exists a
nonempty $G^{c}_{\kappa}$-set $G$ and a set $H\in[X]^{\leq\kappa}$ such that
$G\subseteq\overline{H}$. Then there exists a cover $\mathcal{G}$ of $X$
consisting of $G^{c}_{\kappa}$-sets such that for all $G\in\mathcal{G}$ there
exists $H_{G}\in[X]^{\leq\kappa}$ such that $G\subseteq\overline{H_{G}}$.

Given Theorem LABEL:compactsubset in the next section, one might wonder
whether one can instead suppose $|H|\leq 2^{\kappa}$ in the above corollary
and conclude that each $H_{G}$ has cardinality at most $2^{\kappa}$ while
keeping each $G\in\mathcal{G}$ a $G^{c}_{\kappa}$-set. However, an analysis of
the subtleties of the proof of Corollary LABEL:PH1 does not allow this
adjustment. If each $|H_{i}|\leq 2^{\kappa}$ in Corollary LABEL:PH1 then in
fact the $A$ we obtain from Theorem LABEL:getA2kappa will have cardinality at
most $2^{\kappa}$. In that case, one sees from equation (2.1) in the proof of
Corollary LABEL:PH1 that $\pi^{-1}[G_{A}]$ is a $G^{c}_{2^{\kappa}}$-set, not
necessarily a $G^{c}_{\kappa}$-set. It is essentially for this reason that the
cardinal invariant $at(X)$ is introduced, whereby Theorem
LABEL:atcompactsubset can be used in place of Theorem LABEL:compactsubset in
conjunction with the above corollary. This combination is used in the proof of
Theorem LABEL:cptPH, which gives a cardinality bound for any power homogeneous
compactum.

##  3\. The cardinality of a power homogeneous compactum.

We begin this section by defining the weak tightness $wt(X)$ of a space $X$
and related notions. Given a cardinal $\kappa$, a space $X$, and $A\subseteq
X$, the $\kappa$-_closure_ of $A$ is defined as
$cl_{\kappa}A=\bigcup_{B\in[A]^{\leq\kappa}}\overline{B}$. The definition of
$\kappa$-closure was given in [8] and was used in [13]. Observe that
$cl_{\kappa}(cl_{\kappa}A)=cl_{\kappa}A\subseteq\overline{A}$ and if
$cl_{\kappa}A=X$ then $A$ is dense in $X$.

######  Definition 3.1 ([6]).

Let $X$ be a space. The _weak tightness_ $wt(X)$ of $X$ is defined as the
least infinite cardinal $\kappa$ for which there is a cover $\mathcal{C}$ of
$X$ such that $|\mathcal{C}|\leq 2^{\kappa}$ and for all $C\in\mathcal{C}$,
$t(C)\leq\kappa$ and $X=cl_{2^{\kappa}}C$. We say that $X$ is _weakly
countably tight_ if $wt(X)=\omega$.

######  Definition 3.2.

Let $X$ be a space. The _almost tightness_ $at(X)$ of $X$ is defined as the
least infinite cardinal $\kappa$ for which there is a cover $\mathcal{C}$ of
$X$ such that $|\mathcal{C}|\leq\kappa$ and for all $C\in\mathcal{C}$,
$t(C)\leq\kappa$ and $X=cl_{\kappa}C$. We say that $X$ is _almost countably
tight_ if $at(X)=\omega$.

It is clear that $wt(X)\leq at(X)\leq t(X)$. See Example 2.3 in [4] for a
consistent example of a compact space $X$ for which
$wt(X)=\aleph_{0}<\aleph_{1}=at(X)=t(X)$.

We will see that while $wt(X)$ is used in cardinal inequalities on homogeneous
spaces (Theorems LABEL:cpthomog, LABEL:pbound, LABEL:w(X)), $at(X)$ is more
suited for cardinal inequalities on _power_ homogeneous spaces (Theorem
LABEL:cptPH). This is because $at(X)$ “interacts” well with Corollary
LABEL:GPH. (See comment after that corollary).

In [13], Juhász and van Mill introduced the notion of a
$\mathcal{C}$-saturated subset of a space $X$.

######  Definition 3.3 ([13]).

Given a cover $\mathcal{C}$ of $X$, a subset $A\subseteq X$ is
$\mathcal{C}$-_saturated_ if $A\cap C$ is dense in $A$ for every
$C\in\mathcal{C}$.

It is clear that the union of $\mathcal{C}$-saturated subsets is
$\mathcal{C}$-saturated. The following was mentioned in the proof of Lemma 3.2
in [13] and proved in Proposition 3.2 in [6].

######  Proposition 3.4.

Let $X$ be a space, $wt(X)=\kappa$, and let $\mathcal{C}$ be a cover
witnessing that $wt(X)=\kappa$. Then for all $x\in X$ there exists
$S(x)\in[X]^{\leq 2^{\kappa}}$ such that $x\in S(x)$ and $S(x)$ is
$\mathcal{C}$-saturated.

Whenever $wt(X)=\kappa$ and $x\in X$, we fix $S(x)$ as obtained in Proposition
LABEL:saturated. If $A\subseteq X$, then we set $S(A)=\bigcup_{x\in A}S(x)$.

The following is an adaptation of Proposition LABEL:saturated for the
invariant $at(X)$. Observe that while $S(x)\in[X]^{\leq 2^{\kappa}}$, in fact
$T(x)\in[X]^{\leq\kappa}$. It is this crucial difference that ultimately
allows for the cardinal inequality in Theorem LABEL:cptPH to work for power
homogeneous compacta using $at(X)$ rather than $wt(X)$.

######  Proposition 3.5.

Let $X$ be a space, $at(X)=\kappa$, and let $\mathcal{C}$ be a cover
witnessing that $at(X)=\kappa$. Then for all $x\in X$ there exists
$T(x)\in[X]^{\leq\kappa}$ such that $x\in T(x)$ and $T(x)$ is
$\mathcal{C}$-saturated.

###### Proof.

Fix $x\in X$. We inductively construct a sequence of subsets $A_{n}\subseteq
X$ such that for all $n$, $|A_{n}|\leq\kappa$. For all $C\in\mathcal{C}$ there
exists $A_{C}\in[C]^{\leq\kappa}$ such that $x\in\overline{A_{C}}$. Let
$A_{1}=\bigcup_{C\in\mathcal{C}}A_{C}$ and note that $|A_{1}|\leq\kappa$.
Suppose that $A_{n}$ has been constructed for $n<\omega$. For all $y\in A_{n}$
and for all $C\in\mathcal{C}$ there exists $A(C,y,n)\in[C]^{\leq\kappa}$ such
that $y\in\overline{A(C,y,n)}$. Let $A_{n+1}=\bigcup\\{A(C,y,n):y\in
A_{n},C\in\mathcal{C}\\}$ and note that $|A_{n+1}|\leq\kappa$.

Define $T(x)=\\{x\\}\cup\bigcup_{n<\omega}A_{n}$. Then $x\in T(x)$ and
$|T(x)|\leq\kappa$. We show that $T(x)$ is $\mathcal{C}$-saturated. Let
$C\in\mathcal{C}$. We show $T(x)\subseteq\overline{C\cap T(x)}$. By above,
$x\in\overline{A_{C}}$ and $A_{C}\subseteq C\cap A_{1}\subseteq C\cap T(x)$.
Thus $x\in\overline{C\cap T(x)}$. Now let $n\leq\omega$ and $y\in A_{n}$. By
above, $y\in\overline{A(C,y,n)}$ and $A(C,y,n)\subseteq C\cap A_{n+1}\subseteq
C\cap T(x)$. Thus $y\in\overline{C\cap T(x)}$. This shows that
$T(x)\subseteq\overline{C\cap T(x)}$ and that $T(x)$ is
$\mathcal{C}$-saturated. ∎

Whenever $at(X)=\kappa$ and $x\in X$, we fix $T(x)$ as obtained in Proposition
LABEL:atsaturated. If $A\subseteq X$, then we set $T(A)=\bigcup_{x\in A}T(x)$.

######  Lemma 3.6 (Lemma 2.7 in [6]).

Let $X$ be a space, $\kappa$ a cardinal such that $wt(X)\leq\kappa$, and
$\mathcal{C}$ be a cover of $X$ witnessing that $wt(X)\leq\kappa$. If
$\mathcal{B}$ is an increasing chain of $\kappa^{+}$-many
$\mathcal{C}$-saturated subsets of $X$, then

$\overline{\bigcup\mathcal{B}}=\bigcup_{B\in\mathcal{B}}\overline{B}.$

The notion of an $S$-_free sequence_ was used heavily in connection with the
weak tightness $wt(X)$ in [4]. Recall that for a cardinal $\kappa$ and a space
$X$, a set $\\{x_{\alpha}:\alpha<\kappa\\}\subseteq X$ is a _free sequence_ if
$\overline{\\{x_{\beta}:\beta<\alpha\\}}\cap\overline{\\{x_{\beta}:\alpha\leq\beta<\kappa\\}}=\varnothing$
for all $\alpha<\kappa$. We define the notion of an $S$-free sequence below.
Note that every $S$-free sequence is a free sequence.

######  Definition 3.7 ([4]).

Let $wt(X)=\kappa$. A set $\\{x_{\alpha}:\alpha<\lambda\\}$ is an _S-free
sequence_ if
$\overline{S(\\{x_{\beta}:\beta<\alpha\\})}\cap\overline{\\{x_{\beta}:\alpha\leq\beta<\lambda\\}}=\varnothing$
for all $\alpha<\lambda$.

It was shown in [4] that if $X$ is compact and $wt(X)=\kappa$, then $X$
contains no $S$-free sequences of length $\kappa^{+}$. The following is
related and is a key result in generalizing Theorem LABEL:cpthomog to a non-
compact setting in the next section.

######  Proposition 3.8.

Let $X$ be a space, $K\subseteq X$ be a compact set, and $wt(X)=\kappa$. Then
$K$ contains no $S$-free sequence of length $\kappa^{+}$.

###### Proof.

By way of contradiction suppose $A=\\{x_{\alpha}:\alpha<\kappa^{+}\\}$ is an
$S$-free sequence contained in $K$. As $K$ is compact, $A$ has a complete
accumulation point $x$ in $K$. Thus $|U\cap K\cap A|=\kappa^{+}$ for all open
sets $U$ of $X$ containing $x$. Then, $x\in
cl_{K}A\subseteq\overline{A}\subseteq\overline{S(A)}=\bigcup_{\alpha<\kappa^{+}}\overline{S(\\{x_{\beta}:\beta<\alpha\\})}$,
where the latter equality follows from Lemma LABEL:closureunion and the fact
that $\kappa=wt(X)$. Thus there exists $\alpha<\kappa^{+}$ such that
$x\in\overline{S(\\{x_{\beta}:\beta<\alpha\\})}$. It follows that $x\in
K\backslash\overline{\\{x_{\beta}:\alpha\leq\beta<\kappa^{+}\\}}$, an open set
in $K$. This implies that
$\kappa^{+}=|A\backslash\overline{\\{x_{\beta}:\alpha\leq\beta<\kappa^{+}\\}}|\leq|\\{x_{\beta}:\beta<\alpha\\}|=\kappa$,
a contradiction. ∎

We introduce the notion of a $T$-free sequence for use with the invariant
$at(X)$.

######  Definition 3.9.

Let $at(X)=\kappa$. A set $\\{x_{\alpha}:\alpha<\lambda\\}$ is an _T-free
sequence_ if
$\overline{T(\\{x_{\beta}:\beta<\alpha\\})}\cap\overline{\\{x_{\beta}:\alpha\leq\beta<\lambda\\}}=\varnothing$
for all $\alpha<\lambda$.

It can also be shown that if $K$ is a compact subset of a space $X$ and
$at(X)=\kappa$, then $K$ contains no $T$-free sequence of length $\kappa^{+}$.
The proof is similar to that of Proposition LABEL:noSfree.

######  Proposition 3.10.

Let $X$ be a space, $K\subseteq X$ where $K$ is compact, and $at(X)=\kappa$.
Then $K$ contains no $T$-free sequence of length $\kappa^{+}$.

Arhangel′skiĭ showed in [1] that if $X$ is a compactum and $t(X)\leq\kappa$,
then there exists a nonempty compact set $G$ and a set $H$ such that
$G\subseteq\overline{H}$, $|H|\leq\kappa$ and $\chi(G,X)\leq\kappa$. Juhász
and van Mill [13] improved this in the countable case by showing that if $X$
is a compactum that is the countable union of countably tight subspaces then
there exists a nonempty closed $G_{\delta}$-set contained in the closure of a
countable set. Theorem 3.3 in [4] gave a variation of these results, which
states that if $X$ is a compactum and $wt(X)\leq\kappa$, then there exists a
nonempty compact set $G$ and a $\mathcal{C}$-saturated set $H$ such that
$G\subseteq\overline{H}$, $|H|\leq 2^{\kappa}$, and $\chi(G,X)\leq\kappa$. We
prove an extension of this result into the Hausdorff setting.

######  Theorem 3.11.

Let $X$ be a Hausdorff space, $\kappa=wt(X)$, and $K$ a nonempty compact
subset of $X$. Then there exists a nonempty closed set $G\subseteq K$ and a
set $H\subseteq X$ such that $|H|\leq 2^{\kappa}$, $G\subseteq\overline{H}$,
and $\chi(G,K)\leq\kappa$. In addition, $H$ is $\mathcal{C}$-saturated for any
cover $\mathcal{C}$ witnessing $\kappa=wt(X)$.

###### Proof.

Let $\mathcal{C}$ be a cover of $X$ witnessing that $\kappa=wt(X)$. Set
$\mathcal{F}=\\{F\subseteq K:F\neq\varnothing,F\textup{ closed
},\chi(F,K)\leq\kappa\\}$. Suppose by way of contradiction that for all
$F\in\mathcal{F}$ and all $\mathcal{C}$-saturated $H\in[X]^{\leq 2^{\kappa}}$
we have $F\backslash\overline{H}\neq\varnothing$. We construct a decreasing
sequence
$\mathcal{F}^{\prime}=\\{F_{\alpha}:\alpha<\kappa^{+}\\}\subseteq\mathcal{F}$
and an $S$-free sequence $A=\\{x_{\alpha}:\alpha<\kappa^{+}\\}\subseteq K$.

For $\alpha<\kappa^{+}$, suppose $x_{\beta}\in K$ and $F_{\beta}$ have been
defined for all $\beta<\alpha<\kappa^{+}$, where
$F_{\beta^{\prime\prime}}\subseteq F_{\beta^{\prime}}$ for all
$\beta^{\prime}<\beta^{\prime\prime}<\alpha$. Define
$A_{\alpha}=\\{x_{\beta}:\beta<\alpha\\}$ and
$G_{\alpha}=\bigcap\\{F_{\beta}:\beta<\alpha\\}$. As $K$ is compact,
$G_{\alpha}\neq\varnothing$ and $G_{\alpha}\in\mathcal{F}$. As
$|A_{\alpha}|\leq\kappa$, we have $|S(A_{\alpha})|\leq\kappa\cdot
2^{\kappa}=2^{\kappa}$. Thus,
$G_{\alpha}\backslash\overline{S(A_{\alpha})}\neq\varnothing$ as
$S(A_{\alpha})$ is $\mathcal{C}$-saturated.

Let $x_{\alpha}\in G_{\alpha}\backslash\overline{S(A_{\alpha})}$. As
$\chi(G_{\alpha},K)\leq\kappa$, we have that $G_{\alpha}$ is a
$G_{\kappa}$-set in $K$. There exists a family $\mathcal{U}$ of open sets in
$X$ such that $|\mathcal{U}|\leq\kappa$ and $G_{\alpha}=\bigcap\\{U\cap
K:U\in\mathcal{U}\\}$. As $K$ is regular, for all $U\in\mathcal{U}$ there
exists a closed $G_{\delta}$-set in $K$ such that $x_{\alpha}\in
G_{U}\subseteq(U\cap K)\backslash\overline{S(A_{\alpha})}$, as $(U\cap
K)\backslash\overline{S(A_{\alpha})}$ is open in $K$.

Set $F_{\alpha}=\bigcap\\{G_{U}:U\in\mathcal{U}\\}$. Note $x_{\alpha}\in
F_{\alpha}\subseteq G_{\alpha}\backslash\overline{S(A_{\alpha})}$ and that
$F_{\alpha}$ is a nonempty closed $G_{\kappa}$-set in $K$. As $K$ is compact,
$\chi(F_{\alpha},K)\leq\kappa$ and $F_{\alpha}\in\mathcal{F}$. The choice of
$x_{\alpha}$ and $F_{\alpha}$ completes the construction of the sequences
$\mathcal{F}^{\prime}$ and $A$.

We show now that $A$ is an $S$-free sequence. Let $\alpha<\kappa^{+}$. Note
$\\{x_{\beta}:\alpha\leq\beta<\kappa^{+}\\}\subseteq F_{\alpha}$ as
$\mathcal{F}^{\prime}$ is a decreasing sequence. Then
$\overline{\\{x_{\beta}:\alpha\leq\beta<\kappa^{+}\\}}\subseteq F_{\alpha}$.
Also, $\\{x_{\beta}:\beta<\alpha\\}=A_{\alpha}$, so
$\overline{S(\\{x_{\beta}:\beta<\alpha\\})}=\overline{S(A_{\alpha})}\subseteq
X\backslash F_{\alpha}$. It follows that
$\overline{S(\\{x_{\beta}:\beta<\alpha\\})}\cap\overline{\\{x_{\beta}:\alpha\leq\beta<\kappa^{+}\\}}=\varnothing$
and $A$ is an $S$-free sequence contained in $K$ of length $\kappa^{+}$. This
contradicts Proposition LABEL:noSfree. Thus there exists $G$ and $H$ as
required. ∎

We have a similar result for when $\kappa=at(X)$. The proof is identical,
except that instances of $2^{\kappa}$ are replaced by $\kappa$, a $T$-free
sequence is constructed, and Proposition LABEL:noTfree is used instead of
Proposition LABEL:noSfree.

######  Theorem 3.12.

Let $X$ be a Hausdorff space, $\kappa=at(X)$, and $K$ a nonempty compact
subset of $X$. Then there exists a nonempty closed set $G\subseteq K$ and a
set $H\subseteq X$ such that $|H|\leq\kappa$, $G\subseteq\overline{H}$, and
$\chi(G,K)\leq\kappa$. In addition, $H$ is $\mathcal{C}$-saturated in any
cover $\mathcal{C}$ witnessing that $at(X)=\kappa$.

######  Lemma 3.13 (Lemma 2.4.3 in [17]).

Let $X$ be a space. Suppose $F$ is a compact subset of $Y$ and $Y$ is a
compact subset of $X$. Then $\chi(F,X)\leq\chi(F,Y)\chi(Y,X)$.

######  Definition 3.14.

The _point-wise compactness type_ $pct(X)$ of a space $X$ is the least
infinite cardinal $\kappa$ such that $X$ can be covered by compact sets $K$
such that $\chi(K,X)\leq\kappa$.

Note that if $X$ is compact then $pct(X)=\omega$. Furthermore, it can be shown
that $\chi(X)=\psi(X)pct(X)$.

######  Theorem 3.15.

If $X$ is a Hausdorff space and $wt(X)pct(X)\leq\kappa$, then there exists a
nonempty compact set $G\subseteq X$ and $H\subseteq X$ such that
$\chi(G,X)\leq\kappa$, $G\subseteq\overline{H}$, $|H|\leq 2^{\kappa}$, and $H$
is $\mathcal{C}$-saturated in any cover $\mathcal{C}$ witnessing
$wt(X)\leq\kappa$.

###### Proof.

Since $pct(X)\leq\kappa$, there exists a nonempty compact set $K$ such that
$\chi(K,X)\leq\kappa$. (In fact, there is a cover of such sets). By Theorem
LABEL:compactsubset there exists a nonempty compact set $G\subseteq K$ and a
$\mathcal{C}$-saturated set $H\in[X]^{\leq 2^{\kappa}}$ such that
$G\subseteq\overline{H}$ and $\chi(G,K)\leq\kappa$. Now, by Lemma LABEL:char,
$\chi(G,X)\leq\chi(G,K)\cdot\chi(K,X)=\kappa\cdot\kappa=\kappa$, as $G$ and
$K$ are compact. ∎

We have a related result for when $at(X)pct(X)\leq\kappa$. Notice that
$|H|\leq 2^{\kappa}$ in Theorem LABEL:compactsubset2 and $|H|\leq\kappa$ in
Theorem LABEL:atcompactsubset2. It’s proof is similar to the above and uses
Theorem LABEL:atcompactsubset.

######  Theorem 3.16.

If $X$ is a Hausdorff space and $at(X)pct(X)\leq\kappa$, then there exists a
nonempty compact set $G\subseteq X$ and $H\subseteq X$ such that
$\chi(G,X)\leq\kappa$, $G\subseteq\overline{H}$, $|H|\leq\kappa$, and $H$ is
$\mathcal{C}$-saturated in any cover $\mathcal{C}$ witnessing
$at(X)\leq\kappa$.

The following theorem was established in [6]. It represents an improvement and
generalization of the result of Pytkeev [15] that $L(X_{\kappa})\leq
2^{t(X)\cdot\kappa}$ for compact spaces $X$, where $X_{\kappa}$ is the
$G_{\kappa}$-_modification_ of a space $X$. $X_{\kappa}$ is the space formed
on the underlying set $X$ where the $G_{\kappa}$-sets of $X$ form a basis.
Given a space $X$, $X^{c}_{\kappa}$ represents the
$G^{c}_{\kappa}$-_modification_ of $X$, the space formed on $X$ where the
$G^{c}_{\kappa}$-sets form a basis.

######  Theorem 3.17 (Main Theorem in [6]).

For any space $X$ and cardinal $\kappa$, $L(X^{c}_{\kappa})\leq
2^{L(X)wt(X)\cdot\kappa}$.

######  Lemma 3.18 (Theorem 3.4, [16]).

If $X$ is a power homogeneous Hausdorff space then $|X|\leq
d(X)^{\pi\chi(X)}$.

We now have all the ingredients to prove the central theorem in this section.
It ties together the main result in the previous section with results in this
section.

######  Theorem 3.19.

If $X$ is a power homogeneous compactum then $|X|\leq 2^{at(X)\pi\chi(X)}$.

###### Proof.

Let $\kappa=at(X)\pi\chi(X)$. Let $\mathcal{C}$ be a cover witnessing that
$at(X)\leq\kappa$. By Theorem LABEL:atcompactsubset2 there exists a nonempty
compact set $G$ and a set $H\in[X]^{\leq\kappa}$ such that $H$ is
$\mathcal{C}$-saturated, $G\subseteq\overline{H}$, and $\chi(G,X)\leq\kappa$.
As $G$ is compact, it follows that $G$ is a $G^{c}_{\kappa}$-set. Since $X$ is
power homogeneous and $\pi\chi(X)\leq\kappa$, by Corollary LABEL:GPH there
exists a cover $\mathcal{G}$ of $X$ of nonempty $G^{c}_{\kappa}$-sets such
that for each $G\in\mathcal{G}$ there exists $H_{G}\in[X]^{\leq\kappa}$ such
that $G\subseteq\overline{H_{G}}$. As $wt(X)\leq at(X)$, Theorem LABEL:L(X)
allows us to assume that $|\mathcal{G}|\leq 2^{\kappa}$. Then
$X=\bigcup\mathcal{G}\subseteq\bigcup_{G\in\mathcal{G}}\overline{H_{G}}\subseteq\overline{\bigcup_{G\in\mathcal{G}}H_{G}}$,
showing $D=\bigcup_{G\in\mathcal{G}}H_{G}$ is dense. As $|D|\leq
2^{\kappa}\cdot\kappa=2^{\kappa}$, we have $d(X)\leq 2^{\kappa}$. By Lemma
LABEL:ridPH, it follows that $|X|\leq
d(X)^{\pi\chi(X)}\leq(2^{\kappa})^{\kappa}=2^{\kappa}$, completing the proof.
∎

Theorem LABEL:cptPH improves the result in [2] that the cardinality of a power
homogeneous compactum is at most $2^{t(X)}$, as $\pi\chi(X)\leq t(X)$ for a
compactum $X$ and $at(X)\leq t(X)$ for any space. It also gives a partial
answer to Question 3.13 in [4], which asks if $|X|\leq 2^{wt(X)\pi\chi(X)}$
for a power homogeneous compactum.

Notice that the fact that $\pi\chi(X)\leq\kappa$ is used twice in the above
proof: once in the use of Corollary LABEL:GPH and again in the use of Lemma
LABEL:ridPH. Thus it would seem to be difficult to remove this invariant from
the cardinality bound $2^{at(X)\pi\chi(X)}$ for power homogeneous compacta.

As Theorem LABEL:atcompactsubset2 applies if $at(X)pct(X)\leq\kappa$ and
Theorem LABEL:L(X) applies if $L(X)at(X)\leq\kappa$, observe that the previous
proof in fact demonstrates the following more general result.

######  Theorem 3.20.

Let $X$ be a power homogeneous Hausdorff space. Then $|X|\leq
2^{L(X)at(X)\pi\chi(X)pct(X)}$.

A result in [6] (given in Theorem LABEL:previousPH below) is a consequence of
Theorem LABEL:cptPH. The next several results establish the reasons for this.
Recall that a space is $\sigma$-CT if it is a countable union of countably
tight subspaces. Observe that an almost countably tight space is $\sigma$-CT.
The following was shown by Juhász and van Mill in [13].

######  Theorem 3.21 (Lemma 2.4 in [13]).

If $X$ is a $\sigma$-CT compactum then every nonempty closed subspace $F$ has
a point $x\in F$ such that $\pi\chi(x,F)\leq\omega$.

######  Theorem 3.22 (Lemma 4.2 in [6]).

Let $X$ be a power homogeneous compactum and let $\kappa$ be an infinite
cardinal. If for all nonempty closed subsets $F\subseteq X$ there exists $x\in
F$ such that $\pi\chi(x,F)\leq\kappa$, then $\pi\chi(X)\leq\kappa$.

The following is a direct corollary of the previous two theorems.

######  Corollary 3.23.

Let $X$ be a $\sigma$-CT power homogeneous compactum. Then
$\pi\chi(X)\leq\omega$.

An interesting consequence of Corollary LABEL:pichisigma is the next result.
It follows from the fact that $|X|\leq 2^{c(X)\pi\chi(X)}$ for any power
homogeneous Hausdorff space $X$ [7].

######  Corollary 3.24.

Let $X$ be a $\sigma$-CT power homogeneous compactum. Then $|X|\leq 2^{c(X)}$.

If there is a cover $\mathcal{C}$ of a space $X$ and a cardinal $\kappa$ for
which $|\mathcal{C}|\leq\kappa$ and for all $C\in\mathcal{C}$,
$t(C)\leq\kappa$ and $X=cl_{\kappa}C$ then, by definition, $at(X)\leq\kappa$.
As the next proposition shows, if there is an additional cardinal restriction
on the $\pi$-character $\pi\chi(X)$ then the condition “$X=cl_{\kappa}C$ for
all $C\in\mathcal{C}$ can be relaxed to “$C$ is dense in $X$ for all
$C\in\mathcal{C}$” to still guarantee that $at(X)\leq\kappa$.

######  Proposition 3.25.

Let $X$ be a space, $\kappa$ a cardinal, and $\mathcal{C}$ a cover of $X$ such
that $|\mathcal{C}|\leq\kappa$ and for all $C\in\mathcal{C}$, $t(C)\leq\kappa$
and $C$ is dense in $X$. If $\pi\chi(X)\leq\kappa$ then $at(X)\leq\kappa$.

###### Proof.

It only needs to be shown that $X=cl_{\kappa}(C)$ for all $C\in\mathcal{C}$.
Let $C\in\mathcal{C}$ and let $x\in X$. Let $\mathcal{B}$ be a local
$\pi$-base at $x$ such that $|\mathcal{B}\leq\kappa$. As $C$ is dense in $X$,
for all $B\in\mathcal{B}$ there exists $x_{B}\in B\cap C$. Then
$x\in\overline{\\{x_{B}:B\in\mathcal{B}\\}}$ and $x\in cl_{\kappa}(C)$. ∎

The following is a consequence of Corollary LABEL:pichisigma and Proposition
LABEL:atcondition.

######  Corollary 3.26.

Let $X$ be a power homogeneous compactum and with a countable cover consisting
of dense, countably tight subspaces. Then $X$ is almost countably tight.

The following was proved in [6]. It represents an extension of Theorem 4.1 in
[13]. We see that it is a consequence of Theorem LABEL:cptPH.

######  Theorem 3.27 (Theorem 4.6 [6]).

Let $X$ be a power homogeneous compactum and suppose there exists a countable
cover of $X$ consisting of dense, countably tight subspaces. Then
$|X|\leq\mathfrak{c}$.

###### Proof.

As $X$ is a $\sigma$-CT power homogeneous compactum, by Corollary
LABEL:pichisigma it follows that $\pi\chi(X)\leq\omega$. Now use Corollary
LABEL:almostct to conclude that $X$ is almost countably tight. As $X$ is a
power homogeneous compactum, by Theorem LABEL:cptPH, we have $|X|\leq
2^{at(X)\pi\chi(X)}\leq 2^{\omega\cdot\omega}=\mathfrak{c}$. ∎

##  4\. Generalizations involving the weak tightness.

In this section we extend extend Theorem LABEL:cpthomog (parts (a) and (b)) to
the Hausdorff and regular settings. We will need Theorem LABEL:compactsubset2,
which applies to Hausdorff spaces $X$ with the property
$wt(X)pct(X)\leq\kappa$ for a cardinal $\kappa$. We will also need the
following lemma from [6].

######  Lemma 4.1 (Lemma 3.4 in [6]).

Let $X$ be a space, $D\subseteq X$, $wt(X)\leq\kappa$, and let $\mathcal{G}$
be a cover of $\overline{D}$ consisting of $G^{c}_{\kappa}$-sets of $X$. Then
there exists $\mathcal{G}^{\prime}\subseteq\mathcal{G}$ such that
$|\mathcal{G}^{\prime}|\leq|D|^{\kappa}$ and $\mathcal{G}^{\prime}$ covers
$D$.

The following was defined by Bella and Spadaro in [5].

######  Definition 4.2 ([5]).

Let $X$ be a space. The _piecewise weak Lindelöf degree_ $pwL(X)$ of $X$ is
the least infinite cardinal $\kappa$ such that for every open cover
$\mathcal{U}$ of $X$ and every decomposition $\\{\mathcal{U}_{i}:i\in I\\}$ of
$\mathcal{U}$, there are families
$\mathcal{V}_{i}\in[\mathcal{U}_{i}]^{\leq\kappa}$ for every $i\in I$ such
that $X\subseteq\bigcup\\{\overline{\bigcup\mathcal{V}_{i}}:i\in I\\}$. The
_piecewise weak Lindelöf degree for closed sets_ $pwL_{c}(X)$ of $X$ is the
least infinite cardinal $\kappa$ such that for every closed set $F\subseteq
X$, for every open cover $\mathcal{U}$ of $F$, and every decomposition
$\\{\mathcal{U}_{i}:i\in I\\}$ of $\mathcal{U}$, there are families
$\mathcal{V}_{i}\in[\mathcal{U}_{i}]^{\leq\kappa}$ for every $i\in I$ such
that $F\subseteq\bigcup\\{\overline{\bigcup\mathcal{V}_{i}}:i\in I\\}$.

It is clear that $pwL_{c}(X)\leq L(X)$ and it was shown in [5] that
$pwL_{c}(X)\leq c(X)$. It was also shown in [5] that if $X$ is Hausdorff then
$|X|\leq 2^{pwL_{c}(X)\chi(X)}$, providing a common proof of the well-known
cardinality bounds $2^{L(X)\chi(X)}$ and $2^{c(X)\chi(X)}$ for Hausdorff
spaces.

The following improves Theorem 3.5 in [3] by replacing $t(X)$ with $wt(X)$.
Unlike Theorem 3.5 in [3], the closing-off argument uses an increasing chain
of $\mathcal{C}$-saturated sets.

######  Theorem 4.3.

Let $X$ be a Hausdorff space and $\kappa$ a cardinal such that
$pwL_{c}(X)wt(X)\leq\kappa$. Let $\mathcal{G}$ be a cover of $X$ consisting of
compact sets $G$ such that $\chi(G,X)\leq\kappa$. Then there exists
$\mathcal{G}^{\prime}\subseteq\mathcal{G}$ such that
$X=\overline{\bigcup\mathcal{G}^{\prime}}$ and $|\mathcal{G}^{\prime}|\leq
2^{\kappa}$.

###### Proof.

Let $\mathcal{C}$ be a cover of $X$ witnessing that $wt(X)\leq\kappa$. For all
$G\in\mathcal{G}$, let $\mathcal{U(G)}$ be a neighborhood base at $G$ such
that $|\mathcal{U}(G)|\leq\kappa$. If
$\mathcal{G}^{\prime}\subseteq\mathcal{G}$, let
$\mathcal{U}(\mathcal{G}^{\prime})=\bigcup\\{\mathcal{U}(G):G\in\mathcal{G}^{\prime}\\}$.
Note each $G\in\mathcal{G}$ us a $G^{c}\kappa$-set as each $G\in\mathcal{G}$
is compact, $X$ is Hausdorff, and $\chi(G,X)\leq\kappa$. We build an
increasing chain $\\{A_{\alpha}:\alpha<\kappa^{+}\\}$ of
$\mathcal{C}$-saturated subsets of $X$ and an increasing chain
$\\{\mathcal{G}_{\alpha}:\alpha<\kappa^{+}\\}$ of subsets of $\mathcal{G}$
such that

  1. (1)

$|\mathcal{G}_{\alpha}|\leq 2^{\kappa}$ and $|A_{\alpha}|\leq 2^{\kappa}$,

  2. (2)

$\overline{A_{\alpha}}\subseteq\bigcup\mathcal{G}_{\alpha}$ and each
$A_{\alpha}$ is $\mathcal{C}$-saturated,

  3. (3)

Whenever
$\\{V_{\alpha}:\beta<\kappa\\}\in\left[\left[\mathcal{U}(\mathcal{G}_{\alpha})\right]^{\leq\kappa}\right]^{\leq\kappa}$
for some $\alpha<\kappa^{+}$ and
$X\backslash\bigcup\\{\overline{\bigcup\mathcal{V}_{\beta}}:\beta<\kappa\\}\neq\varnothing$,
then
$A_{\alpha+1}\backslash\bigcup\\{\overline{\bigcup\mathcal{V}_{\beta}}:\beta<\kappa\\}\neq\varnothing$.

Let $A_{0}$ be an arbitrary $\mathcal{C}$-saturated set of cardinality at most
$2^{\kappa}$. (For example, let $A_{0}=S(x)$ for some $x\in X$). For limit
ordinals $\alpha<\kappa^{+}$, let
$A_{\alpha}=\bigcup_{\beta<\alpha}A_{\beta}$. Then $|A_{\alpha}|\leq
2^{\kappa}$, and $A_{\alpha}$ is $\mathcal{C}$-saturated as is a union of
$\mathcal{C}$-saturated sets is $\mathcal{C}$-saturated. By applying Lemma
LABEL:subcover to $\overline{A_{\alpha}}$, we get $\mathcal{G}_{\alpha}$ as
required.

For successor ordinals $\alpha+1$, we ensure (3) above is satisfied. For all
$\mathcal{V}=\\{\mathcal{V}_{\beta}:\beta<\kappa\\}\in\left[\left[\mathcal{U}(\mathcal{G}_{\alpha})\right]^{\leq\kappa}\right]^{\leq\kappa}$
such that
$X\backslash\bigcup\\{\overline{\bigcup\mathcal{V}_{\beta}}:\beta<\kappa\\}\neq\varnothing$,
let $x_{\mathcal{V}}\in
X\backslash\bigcup\\{\overline{\bigcup\mathcal{V}_{\beta}}:\beta<\kappa\\}$.
Define

$\displaystyle A_{\alpha+1}$
$\displaystyle=A_{\alpha}\cup\\{S(x_{\mathcal{V}}):\mathcal{V}=\\{\mathcal{V}_{\beta}:\beta<\kappa\\}\in\left[\left[\mathcal{U}(\mathcal{G}_{\alpha})\right]^{\leq\kappa}\right]^{\leq\kappa}\textup{
and }$ $\displaystyle
X\backslash\bigcup\\{\overline{\bigcup\mathcal{V}_{\beta}}:\beta<\kappa\\}\neq\varnothing\\}$

As $|S(x_{\mathcal{V}})|\leq 2^{\kappa}$ and
$\left|\left[\left[\mathcal{U}(\mathcal{G}_{\alpha})\right]^{\leq\kappa}\right]^{\leq\kappa}\right|\leq
2^{\kappa}$, we have $|A_{\alpha+1}|\leq 2^{\kappa}$. As each
$S(x_{\mathcal{V}})$ is $\mathcal{C}$-saturated, it follows that
$A_{\alpha+1}$ is $\mathcal{C}$-saturated. Apply Lemma LABEL:subcover again to
$\overline{A_{\alpha+1}}$ to obtain $\mathcal{G}_{\alpha+1}$.

Let $A=\bigcup\\{A_{\alpha}:\alpha<\kappa^{+}\\}$. By Lemma
LABEL:closureunion, it follows that
$\overline{A}=\bigcup\\{\overline{A_{\alpha}}:\alpha<\kappa^{+}\\}$. Let
$\mathcal{G}^{\prime}=\bigcup\\{\mathcal{G}_{\alpha}:\alpha<\kappa^{+}\\}$ and
note that $\overline{A}\subseteq\bigcup\mathcal{G}^{\prime}$. We show
$X=\overline{\bigcup\mathcal{G}^{\prime}}$. Suppose by way of contradiction
that $X\backslash\overline{\bigcup\mathcal{G}^{\prime}}\neq\varnothing$ and
let $x\in X\backslash\overline{\bigcup\mathcal{G}^{\prime}}$. Now there exists
$G\in\mathcal{G}$ such that $x\in G$, as $\mathcal{G}$ is a cover of $X$.
Since $G$ is compact and $X$ is Hausdorff, it follows that $G$ is regular.
Note that $x\in G\backslash\overline{\bigcup\mathcal{G}^{\prime}}$ and that
$G\backslash\overline{\bigcup\mathcal{G}^{\prime}}$ is an open set in $G$. By
regularity of $G$ there exists a set $H\subseteq G$ such that $H$ is a closed
$G_{\delta}^{c}$-set in $G$ and $x\in H\subseteq
G\backslash\overline{\bigcup\mathcal{G}^{\prime}}$. Observe that $H$ is
compact as it is a closed subset of the compact set $G$. Thus
$\omega=\psi(H,G)=\chi(H,G)$. Then, by Lemma LABEL:char,
$\chi(H,X)\leq\chi(H,G)\cdot\chi(G,X)\leq\omega\cdot\kappa=\kappa$ because $H$
and $G$ are compact.

Let $\mathcal{N}$ be a neighborhood base at $H$ in $X$ such that
$|\mathcal{N}|\leq\kappa$. Now, for all $G\in\mathcal{G}^{\prime}$ we have
$H\cap G=\varnothing$. Since $H$ and $G$ are compact and $X$ is Hausdorff,
there exists $N_{G}\in\mathcal{N}$ and $U(G)\in\mathcal{U}(G)$ such that
$N_{G}\cap U(G)=\varnothing$. Let
$\mathcal{U}=\\{U(G):G\in\mathcal{G}^{\prime}\\}$ and note $\mathcal{U}$ is a
cover of $\overline{A}$. For all $N\in\mathcal{N}$ let
$\mathcal{U}_{N}=\\{U\in\mathcal{U}:U\cap N=\varnothing\\}$ and observe that
$x\in H\subseteq X\backslash\overline{\bigcup\mathcal{U}_{N}}$. The family
$\\{\mathcal{U}_{N}:N\in\mathcal{N}\\}$ is a decomposition of $\mathcal{U}$.
Since $pwL_{c}(X)\leq\kappa$, for all $N\in\mathcal{N}$ there exists
$\mathcal{V}_{N}\in[\mathcal{U}_{N}]^{\leq\kappa}$ such that
$\overline{A}\subseteq\bigcup\\{\overline{\bigcup\mathcal{V}_{N}}:N\in\mathcal{N}\\}$.
Then $x\in H\subseteq X\backslash\overline{\bigcup\mathcal{V}_{N}}$ for each
$N\in\mathcal{N}$ and $x\in
X\backslash\bigcup\\{\overline{\bigcup\mathcal{V}_{N}}:N\in\mathcal{N}\\}\neq\varnothing$.
As $\\{\mathcal{V}_{N}:N\in\mathcal{N}\\}$ fits condition (3) above and
$\\{\mathcal{V}_{N}:N\in\mathcal{N}\\}\in\left[\left[\mathcal{U}(\mathcal{G}_{\alpha})\right]^{\leq\kappa}\right]^{\leq\kappa}$
for some $\alpha<\kappa^{+}$, we have

$\varnothing\neq
A_{\alpha+1}\backslash\bigcup\\{\overline{\bigcup\mathcal{V}_{N}}:N\in\mathcal{N}\\}\subseteq
A\subseteq\overline{A}\subseteq\bigcup\\{\overline{\bigcup\mathcal{V}_{N}}:N\in\mathcal{N}\\},$

a contradiction. Thus $X=\overline{\bigcup\mathcal{G}^{\prime}}$ which
completes the proof. ∎

If $X$ is homogeneous the above theorem is used to obtain a bound for the
cardinality of $X$.

######  Theorem 4.4.

If $X$ is homogeneous and Hausdorff then $|X|\leq
2^{pwL_{c}(X)wt(X)\pi\chi(X)pct(X)}$.

###### Proof.

Let $\kappa=pwL_{c}(X)wt(X)\pi\chi(X)pct(X)$, and let $\mathcal{C}$ be a cover
of $X$ witnessing that $wt(X)\leq\kappa$. As $wt(X)pct(X)\leq\kappa$, by
Theorem LABEL:compactsubset2 there exists a nonempty compact set $G\subseteq
X$ and $H\subseteq X$ such that $|H|\leq 2^{\kappa}$,
$G\subseteq\overline{H}$, $\chi(G,X)\leq\kappa$, and $H$ is
$\mathcal{C}$-saturated.

Fix $p\in X$. For all $x\in X$ there exists a homeomorphism $h_{x}:X\to X$
such that $h_{x}(p)=X$. Then $\mathcal{G}^{\prime}=\\{h_{x}[G]:x\in X\\}$ is a
cover of $X$ by compact sets of character at most $\kappa$. By Theorem
LABEL:pwL_c above there exists a subfamily
$\mathcal{G}\subseteq\mathcal{G}^{\prime}$ such that $|\mathcal{G}|\leq
2^{\kappa}$ and $X=\overline{\bigcup\mathcal{G}}$. Also, for all
$G\in\mathcal{G}$ there exists $H_{G}\in[X]^{\leq 2^{\kappa}}$ such that
$G\subseteq\overline{H_{G}}$. It follows that
$D=\bigcup_{G\in\mathcal{G}}H_{G}$ is dense in $X$. Since $|D|\leq
2^{\kappa}\cdot 2^{\kappa}=2^{\kappa}$, we have $d(X)\leq 2^{\kappa}$. Now
apply the fact that $|X|\leq d(X)^{\pi\chi(X)}$ for any Hausdorff homogeneous
space (Lemma LABEL:ridPH). ∎

Despite the fact that four cardinal invariants are used in the bound in
Theorem LABEL:pbound, each of these invariants should be regarded as “small”
in a rough sense. In addition, as $\pi\chi(X)\leq t(X)pct(X)$ for a Hausdorff
space $X$ and $wt(X)\leq t(X)$, we see that Theorem LABEL:pbound improves
Corollary 3.7 in [3]. This latter result states that if $X$ is homogeneous and
Hausdorff, then $|X|\leq 2^{pwL_{c}(X)t(X)pct(X)}$. Theorem LABEL:pbound is
also an extension of Theorem LABEL:cpthomog(b).

It was shown in [7] that the cardinality of a homogeneous Hausdorff space $X$
is at most $2^{c(X)\pi\chi(X)}$. Observe that the above bound is a variation
of this, as $pwL_{c}(X)\leq c(X)$ and $wt(X)$ and $pct(X)$ are added to the
bound in Theorem LABEL:pbound.

######  Lemma 4.5 (Lemma 3.4.8 in [17]).

Let $X$ be a space, and suppose $pct(X)\leq\kappa$ and $nw(X)\leq 2^{\kappa}$.
Then $w(X)\leq 2^{\kappa}$.

######  Corollary 4.6.

If $X$ is homogeneous and Hausdorff then $w(X)\leq
2^{pwL_{c}(X)wt(X)\pi\chi(X)pct(X)}$.

###### Proof.

Let $\kappa=pwL_{c}(X)wt(X)\pi\chi(X)pct(X)$. By Theorem LABEL:pbound, we have
$nw(X)\leq|X|\leq 2^{\kappa}$. As $pct(X)\leq\kappa$, it follows by Lemma
LABEL:nw2w that $w(X)\leq 2^{\kappa}$. ∎

######  Lemma 4.7 (Proposition 3.5 in [4]).

Let $X$ be a regular space, $\kappa=wt(X)$, and $D\subseteq X$ be
$\mathcal{C}$-saturated. Then $nw(\overline{D})\leq|D|^{\kappa}$.

We arrive at a result for regular, homogeneous spaces. It extends Theorem
LABEL:cpthomog(a), which states that $w(X)\leq 2^{wt(X)}$ for homogeneous
compacta. Compare to Corollary LABEL:anotherw(X).

######  Theorem 4.8.

Let $X$ be a homogeneous regular Hausdorff space. Then $w(X)\leq
2^{L(X)wt(X)pct(X)}$.

###### Proof.

Let $\kappa=L(X)wt(X)pct(X)$, and let $\mathcal{C}$ be a cover of $X$
witnessing that $wt(X)\leq\kappa$. By Theorem LABEL:compactsubset2, there
exists a nonempty compact set $G\subseteq X$ and $H\subseteq X$ such that $H$
is $\mathcal{C}$-saturated, $|H|\leq 2^{\kappa}$, $G\subseteq\overline{H}$,
and $\chi(G,X)\leq\kappa$. Fix $p\in G$. As $X$ is homogeneous, for all $x\in
X$ there exists a homeomorphism $h_{x}:X\to X$ such that $h_{x}(p)=x$. Then
$\mathcal{F}=\\{h_{x}[G]:x\in X\\}$ is a cover of $X$ by compact
$G^{c}_{\kappa}$-sets and, for all $x\in X$, $h_{x}[G]\subseteq
h_{x}[\overline{H}]$. By Theorem LABEL:L(X), there exists $A\subseteq X$ such
that $|A|\leq 2^{\kappa}$ and $X=\bigcup\\{h_{x}[G]:x\in
A\\}=\bigcup\\{h_{x}[\overline{H}]:x\in A\\}$.

As $H$ is $\mathcal{C}$-saturated and $X$ is regular, by Lemma
LABEL:Csatnetwork it follows that
$nw(\overline{H})\leq|H|^{\kappa}\leq(2^{\kappa})^{\kappa}=2^{\kappa}$. Let
$\mathcal{N}$ be a network for $\overline{H}$ such that $|\mathcal{N}|\leq
2^{\kappa}$. Let $\mathcal{M}=\\{h_{x}[N]:N\in\mathcal{N},x\in A\\}$ and note
$|\mathcal{M}|\leq 2^{\kappa}\cdot 2^{\kappa}=2^{\kappa}$. We show
$\mathcal{M}$ is a network for $X$. Let $y\in U$ where $U$ is open in $X$.
There exists $x\in A$ such that $y\in h_{x}[\overline{H}]\cap U$. Then
$h_{x}^{\leftarrow}(y)\in\overline{H}\cap h_{x}^{\leftarrow}[U]$. As
$\mathcal{N}$ is a network for $\overline{H}$, there exists $N\in\mathcal{N}$
such that $h_{x}^{\leftarrow}(y)\in N\subseteq\overline{H}\cap
h_{x}^{\leftarrow}[U]$. It follows that $y\in h_{x}[N]\subseteq
h_{x}[\overline{H}]\cap U\subseteq U$. As $h_{x}[N]\in\mathcal{M}$, we have
that $\mathcal{M}$ is a network for $X$ and $nw(X)\leq 2^{\kappa}$. Therefore,
as $pct(X)\leq\kappa$, it follows by Lemma LABEL:nw2w that $w(X)\leq
2^{\kappa}$. This completes the proof. ∎

We do not know if Theorem LABEL:w(X) holds when the homogeneous condition is
replaced with the weaker power homogeneous property.

######  Question 4.9.

If $X$ is a power homogeneous regular Hausdorff space, is $w(X)\leq
2^{L(X)wt(X)pct(X)}$?

######  Question 4.10.

If $X$ is power homogeneous and Hausdorff, is $|X|\leq
2^{pwL_{c}(X)wt(X)\pi\chi(X)pct(X)}$?

Finally, as Theorem LABEL:cptPH only gives a partial answer to Question 3.13
in [5], we ask that question again.

######  Question 4.11.

If $X$ is a power homogeneous compactum, is $|X|\leq 2^{wt(X)\pi\chi(X)}$?

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