$B_{\alpha}$ is replaced by $A$). Notice that $\mathscr{A}$ is trivially
closed under complementation. We first show that $\mathscr{A}$ contains all
closed subsets of $X$.

######  Claim 4.1.1.

Let $C$ be a closed subset of an $f\mathrm{SC}_{\kappa}$-space $(X,\tau)$.
Then there is a topology $\tau^{\prime}$ which satisfies (1)–(4) above (where
in (2) the set $B_{\alpha}$ is replaced by $C$).

###### Proof of the Claim.

Let $\bar{\tau}$ be the smallest topology generated by $\tau\cup\\{C\\}$. Then
(1)–(3) are trivially satisfied. Furthermore, $(X,\bar{\tau})$ is homeomorphic
to the sum of the spaces $C$ and $X\setminus C$ (endowed with the relative
topologies inherited from $X$). Since both $C$ and $X\setminus C$ are
$f\mathrm{SC}_{\kappa}$-spaces by Theorem 2.16, and since the class of
$f\mathrm{SC}_{\kappa}$-spaces is trivially closed under ($\leq\kappa$-sized)
sums, then $X$ is an $f\mathrm{SC}_{\kappa}$-space as well. Applying
Proposition 2.33 to $(X,\bar{\tau})$ we then get a topology
$\tau^{\prime}\supseteq\bar{\tau}\supseteq\tau$ which satisfies all of
(1)–(4). ∎

######  Claim 4.1.2.

Let $(A_{\alpha})_{\alpha<\kappa}$ be a family of sets in $\mathscr{A}$. Then
there is a topology $\tau^{\prime}_{\infty}$ simultaneously witnessing
$A_{\alpha}\in\mathscr{A}$ for all $\alpha<\kappa$.

###### Proof of the Claim.

For each $\alpha<\kappa$ let $\tau^{\prime}_{\alpha}$ be a topology witnessing
$A_{\alpha}\in\mathscr{A}$. Define $\tau^{\prime}_{\infty}$ as the smallest
$\kappa$-additive topology containing
$\bigcup_{\alpha<\kappa}\tau^{\prime}_{\alpha}$. Then (1)–(3) are obvious,
since $\tau^{\prime}_{\infty}$ refines each
$\tau^{\prime}_{\alpha}\supseteq\tau$ and
$\tau^{\prime}_{\infty}\subseteq\mathsf{Bor}_{\kappa}(X,\tau)$. To prove (4),
for each $\alpha<\kappa$ fix a closed
$C_{\alpha}\subseteq\tensor[^{\kappa}]{\kappa}{}$ and a homeomorphism
$h_{\alpha}\colon C_{\alpha}\to(X,\tau^{\prime}_{\alpha})$ as given by Theorem
2.16. Endow ${}^{\kappa}(\tensor[^{\kappa}]{\kappa}{})$ with the
$\kappa$-supported product topology, i.e. the topology generated by the sets
$\prod_{\alpha<\kappa}U_{\alpha}$, where each $U_{\alpha}$ is open in the
bounded topology of $\tensor[^{\kappa}]{\kappa}{}$, and only ${<}\kappa$-many
of them differ from $\tensor[^{\kappa}]{\kappa}{}$. Then
$\prod_{\alpha<\kappa}C_{\alpha}$ is closed in
${}^{\kappa}(\tensor[^{\kappa}]{\kappa}{})$, and since the maps $h_{\alpha}$
are continuous, the set

$\Delta=\left\\{(x_{\alpha})_{\alpha<\kappa}\in\prod_{\alpha<\kappa}C_{\alpha}\Bigm{|}\forall\alpha,\beta<\kappa\,\Big{(}h_{\alpha}(x_{\alpha})=h_{\beta}(x_{\beta})\Big{)}\right\\}$

is closed as well. It is then easy to check that the map
$h\colon\Delta\to(\tensor[^{\kappa}]{\kappa}{},\tau^{\prime}_{\infty})$
sending $(x_{\alpha})_{\alpha<\kappa}\in\Delta$ to $h_{0}(x_{0})$ is a
homeomorphism. Therefore the desired result follows from Theorem 2.16 and the
fact that the spaces ${}^{\kappa}(\tensor[^{\kappa}]{\kappa}{})$ and
$\tensor[^{\kappa}]{\kappa}{}$ are clearly homeomorphic. ∎

Claim 4.1.2 in particular reduces our task of proving the theorem for a whole
family $(B_{\alpha})_{\alpha<\kappa}$ to showing that $B\in\mathscr{A}$ for
every single $\kappa$-Borel set $B\subseteq X$. To this aim, by Claim 4.1.1
and closure of $\mathscr{A}$ under complementation it is enough to show that
$\mathscr{A}$ is closed under intersections of length $\leq\kappa$. So let
$A=\bigcap_{\alpha<\kappa}A_{\alpha}$ be such that $A_{\alpha}\in\mathscr{A}$
for every $\alpha<\kappa$. By Claim 4.1.2, there is a topology
$\tau^{\prime}_{\infty}$ simultaneously witnessing $A_{\alpha}\in\mathscr{A}$
for all $\alpha<\kappa$. Then $A$ is closed in the $\kappa$-additive
$f\mathrm{SC}_{\kappa}$-space $(X,\tau^{\prime}_{\infty})$. Therefore Claim
4.1.1 applied to $A$, viewed as a subset of $(X,\tau^{\prime}_{\infty})$,
yields the desired topology
$\tau^{\prime}\supseteq\tau^{\prime}_{\infty}\supseteq\tau$. ∎

Proposition 4.1 provides an alternative proof of [LS15, Lemma 1.11]: Every
$\kappa$-Borel subset of $\tensor[^{\kappa}]{\kappa}{}$ equals a continuous
injective image of a closed subset of $\tensor[^{\kappa}]{\kappa}{}$. To see
this, let $B\subseteq\tensor[^{\kappa}]{\kappa}{}$ be $\kappa$-Borel, and let
$\tau^{\prime}$ be the topology obtained by applying Proposition 4.1 with
$B_{\alpha}=B$ for all $\alpha<\kappa$. Let $D$ be a closed subset of
$\tensor[^{\kappa}]{\kappa}{}$ and
$h\colon(D,\tau_{b})\to(\tensor[^{\kappa}]{\kappa}{},\tau^{\prime})$ be a
homeomorphism as given by Proposition 2.13. Then $C=h^{-1}(B)$ is closed in
$D$ and hence in $\tensor[^{\kappa}]{\kappa}{}$. Moreover, the map
$h^{\prime}\colon(D,\tau_{b})\to(\tensor[^{\kappa}]{\kappa}{},\tau_{b})$
obtained by composing $h$ with the identity function
$(\tensor[^{\kappa}]{\kappa}{},\tau^{\prime})\to(\tensor[^{\kappa}]{\kappa}{},\tau_{b})$
is still a continuous bijection because $\tau^{\prime}\supseteq\tau_{b}$.
Therefore, $h^{\prime}\restriction C$ is a continuous injection from the
closed set $C\subseteq\tensor[^{\kappa}]{\kappa}{}$ onto $B$. Notice also
that, by construction, $h^{\prime}$ is actually a $\kappa$-Borel isomorphism
because
$\mathsf{Bor}_{\kappa}(\tensor[^{\kappa}]{\kappa}{},\tau^{\prime})=\mathsf{Bor}_{\kappa}(\tensor[^{\kappa}]{\kappa}{},\tau_{b})$.
More generally, the same argument shows that [LS15, Lemma 1.11] can be
extended to arbitrary $f\mathrm{SC}_{\kappa}$-spaces.

######  Corollary 4.2.

For every $\kappa$-Borel subset $B$ of an $f\mathrm{SC}_{\kappa}$-space there
is a continuous $\kappa$-Borel isomorphism from a closed
$C\subseteq\tensor[^{\kappa}]{\kappa}{}$ to $B$.

The space $C$ in the previous corollary is an $f\mathrm{SC}_{\kappa}$-space by
Theorem 2.16, hence applying Theorem 2.38 we further get

######  Corollary 4.3.

Each $\kappa$-Borel subset $B$ of an $f\mathrm{SC}_{\kappa}$-space is
$\kappa$-Borel isomorphic to a $\kappa$-additive $\mathrm{SC}_{\kappa}$-space.

The following is the counterpart of Proposition 4.1 in terms of functions and
can be proved by applying it to the preimages of the open sets in any
$\leq\kappa$-sized basis for the topology of $Y$.

######  Corollary 4.4.

Let $(X,\tau)$ be an $f\mathrm{SC}_{\kappa}$-space and $Y$ be any space of
weight $\leq\kappa$. Then for every $\kappa$-Borel function $f\colon X\to Y$
there is a topology $\tau^{\prime}$ on $X$ such that:

  1. (1)

$\tau^{\prime}$ refines $\tau$;

  2. (2)

$f\colon(X,\tau^{\prime})\to Y$ is continuous,

  3. (3)

$\mathsf{Bor}_{\kappa}(X,\tau^{\prime})=\mathsf{Bor}_{\kappa}(X,\tau)$, and

  4. (4)

$(X,\tau^{\prime})$ is a $\kappa$-additive $f\mathrm{SC}_{\kappa}$-space.

Finally, combining the results obtained so far we get that all the proposed
generalizations of (St.Bor. 1) and (St.Bor. 2) give rise to the same class of
spaces. In particular, up to $\kappa$-Borel isomorphism such class coincide
with any of the classes of Polish-like spaces we analyzed in the previous
sections. (Notice also that Theorem 4.5 substantially strengthens [CS16,
Corollary 3.4].)

######  Theorem 4.5.

A $\kappa$-Borel space $(X,\mathscr{B})$ is standard if and only if there is a
topology $\tau$ on $X$ such that

  1. (1)

$(X,\tau)$ is an $f\mathrm{SC}_{\kappa}$-space, and

  2. (2)

$\mathsf{Bor}_{\kappa}(X,\tau)=\mathscr{B}$.

Moreover, condition (1) can equivalently be replaced by

  1. (1′)

$(X,\tau)$ is a $\kappa$-additive $\mathrm{SC}_{\kappa}$-space.

######  Remark 4.6.

Since $\kappa$-additive $\mathrm{SC}_{\kappa}$-spaces and
$f\mathrm{SC}_{\kappa}$-spaces form, respectively, the smallest and largest
class of Polish-like spaces considered in this paper, in Theorem 4.5 we can
further replace those classes with any of the other ones: $\kappa$-additive
$f\mathrm{SC}_{\kappa}$-spaces, $\mathrm{SC}_{\kappa}$-spaces,
$\mathbb{G}$-Polish spaces, spherically complete $\mathbb{G}$-Polish spaces,
and so on.

A natural question is to ask whether Proposition 4.1 can be extended in at
least some direction. As in the classical case, the answer is mostly negative
and thus Proposition 4.1 is essentially optimal. In fact:

  1. (a)

We cannot in general consider more than $\kappa$-many (even closed, or open)
subsets, since this could force $\tau^{\prime}$ to have weight greater than
$\kappa$—think about turning into clopen sets more than $\kappa$-many
singletons.

  2. (b)

We obviously cannot turn a set which is not $\kappa$-Borel into a clopen (or
even just $\kappa$-Borel) one pretending to maintain the same $\kappa$-Borel
structure. Notice however that, in contrast to the classical case, one can
consistently have that there are non-$\kappa$-Borel sets
$B\subseteq\tensor[^{\kappa}]{\kappa}{}$ for which there is a
$\kappa$-additive $f\mathrm{SC}_{\kappa}$ topology
$\tau^{\prime}\supseteq\tau_{b}$ turning $B$ into a $\tau^{\prime}$-clopen
set, so that all conditions in Proposition 4.1 except for (3) are satisfied
with respect to such $B$ (see Corollary 5.10 for more details and
limitations).

  3. (c)

By Example 2.35, we cannot require that the topology $\tau^{\prime}$ be
$\mathrm{SC}_{\kappa}$ (instead of just $f\mathrm{SC}_{\kappa}$). The same
remains true if we consider a single $\kappa$-Borel set $B$ (instead of a
whole family $(B_{\alpha})_{\alpha<\kappa}$), we start from the stronger
hypothesis that $(X,\tau)$ is already a $\kappa$-additive
$\mathrm{SC}_{\kappa}$-space, and we weaken the conclusions by dropping
condition (3) and relaxing condition (2) to “$B$ is $\tau^{\prime}$-open” (or
“$B$ is $\tau^{\prime}$-closed”).

As it is clear from the discussion, in the last item the problem arises from
the fact that there is a tension between condition (1) and our desire to
strengthen condition (4) from $f\mathrm{SC}_{\kappa}$ to
$\mathrm{SC}_{\kappa}$. However, we are now going to show that if we drop the
problematic condition (1), then it is possible to obtain the desired
strengthening, at least when we just consider a few $\kappa$-Borel sets at a
time.

######  Proposition 4.7.

For every $\kappa$-Borel subset $B$ of an $f\mathrm{SC}_{\kappa}$-space
$(X,\tau)$ there is a topology $\tau^{\prime\prime}$ on $X$ such that:

  1. (1)

$B$ is $\tau^{\prime\prime}$-clopen,

  2. (2)

$\mathsf{Bor}_{\kappa}(X,\tau^{\prime\prime})=\mathsf{Bor}_{\kappa}(X,\tau)$,
and

  3. (3)

$(X,\tau^{\prime\prime})$ is a $\kappa$-additive $\mathrm{SC}_{\kappa}$-space
(hence so are its subspaces $B$ and $X\setminus B$ because they are
$\tau^{\prime\prime}$-open).

###### Proof.

By Corollary 4.3, there are $\kappa$-additive $\mathrm{SC}_{\kappa}$
topologies $\tau_{1}$ and $\tau_{2}$ on, respectively, $B$ and $X\setminus B$
such that
$\mathsf{Bor}_{\kappa}(B,\tau_{1})=\mathsf{Bor}_{\kappa}(X,\tau)\restriction
B$ and $\mathsf{Bor}_{\kappa}(X\setminus
B,\tau_{2})=\mathsf{Bor}_{\kappa}(X,\tau)\restriction(X\setminus B)$. Let
$\tau^{\prime\prime}$ be the topology on $X$ construed as the sum of
$(B,\tau_{1})$ and $(X\setminus B,\tau_{2})$: then $\tau^{\prime\prime}$ is as
required. ∎

The proof of Proposition 4.7 can easily be adapted to work with $\kappa$-many
_pairwise disjoint_ $\kappa$-Borel subsets of $X$. This in turn implies that
the proposition can e.g. be extended to deal with ${<}\kappa$-many
$\kappa$-Borel sets simultaneously, even when such sets are not pairwise
disjoint. Indeed, if $(B_{\alpha})_{\alpha<\nu}$ with $\nu<\kappa$ is such a
family, then for each $s\in\tensor[^{\nu}]{2}{}$ we can set

$C_{s}=\\{x\in X\mid\forall\alpha<\nu\,(x\in B_{\alpha}\iff s(\alpha)=1)\\}.$

Since $2^{\nu}\leq\kappa^{<\kappa}=\kappa$, the family
$(C_{s})_{s\in\tensor[^{\nu}]{2}{}}$ is a partition of $X$ into
$\leq\kappa$-many $\kappa$-Borel sets, and any topology $\tau^{\prime\prime}$
working simultaneously for all the $C_{s}$ will work for all sets in the
family $(B_{\alpha})_{\alpha<\nu}$ as well. In contrast, Proposition 4.7
cannot be extended to arbitrary $\kappa$-sized families of $\kappa$-Borel
sets, even when we restrict to $X=\tensor[^{\kappa}]{\kappa}{}$. Indeed, let
$C\subseteq\tensor[^{\kappa}]{\kappa}{}$ be as in Example 2.35 and let
$(B_{\alpha})_{\alpha<\kappa}$ be an enumeration of
$\\{C\\}\cup\\{\mathbf{N}_{s}\cap C\mid s\in\tensor[^{<\kappa}]{\kappa}{}\\}$.
Then $(B_{\alpha})_{\alpha<\kappa}$ is a family of Borel subsets of
$\tensor[^{\kappa}]{\kappa}{}$ such that there is no $\mathrm{SC}_{\kappa}$
topology $\tau^{\prime\prime}$ on $\tensor[^{\kappa}]{\kappa}{}$ making each
$B_{\alpha}$ a $\tau^{\prime\prime}$-open subset of
$\tensor[^{\kappa}]{\kappa}{}$, since otherwise
$\tau^{\prime\prime}\restriction C$ would be an $\mathrm{SC}_{\kappa}$
topology on $C$ refining $\tau_{b}\restriction C$.

From a different perspective, it might be interesting to understand which
subspaces of a Polish-like space inherit a standard $\kappa$-Borel structure
form their superspace. Of course this includes all $\kappa$-Borel sets, as
standard $\kappa$-Borel spaces are closed under $\kappa$-Borel subspaces, and
we are now going to show that no other set has such property. We begin with a
preliminary result which is of independent interest, as it shows that if a
(regular Hausdorff) topology of weight $\leq\kappa$ induces a standard
$\kappa$-Borel structure, then it can be refined to a Polish-like topology
with the same $\kappa$-Borel sets.

######  Proposition 4.8.

Let $(X,\tau)$ be a space of weight $\leq\kappa$. We have that
$(X,\mathsf{Bor}_{\kappa}(X,\tau))$ is a standard $\kappa$-Borel space if and
only if there is a topology $\tau^{\prime}\supseteq\tau$ such that
$(X,\tau^{\prime})$ is a $\kappa$-additive $f\mathrm{SC}_{\kappa}$-space and
$\mathsf{Bor}_{\kappa}(X,\tau)=\mathsf{Bor}_{\kappa}(X,\tau^{\prime})$.

###### Proof.

The backward implication follows from Theorem 4.5. For the forward
implication, suppose that $(X,\mathsf{Bor}_{\kappa}(X,\tau))$ is standard
$\kappa$-Borel. By Theorem 4.5, there is a topology $\hat{\tau}$ such that
$(X,\hat{\tau})$ is an $f\mathrm{SC}_{\kappa}$-space, and
$\mathsf{Bor}_{\kappa}(X,\hat{\tau})=\mathsf{Bor}_{\kappa}(X,\tau)$. Then the
identity function $i\colon(X,\hat{\tau})\to(X,\tau)$ satisfies the hypothesis
of Corollary 4.4, hence there is a $\kappa$-additive $f\mathrm{SC}_{\kappa}$
topology $\tau^{\prime}$ such that $i\colon(X,\tau^{\prime})\to(X,\tau)$ is
continuous and
$\mathsf{Bor}_{\kappa}(X,\tau^{\prime})=\mathsf{Bor}_{\kappa}(X,\hat{\tau})=\mathsf{Bor}_{\kappa}(X,\tau)$,
which implies $\tau\subseteq\tau^{\prime}$. ∎

######  Theorem 4.9.

Let $(X,\mathscr{B})$ be a standard $\kappa$-Borel space, and let $A\subseteq
X$. Then $(A,\mathscr{B}\restriction A)$ is a standard $\kappa$-Borel space if
and only if $A\in\mathscr{B}$.

###### Proof.

Since $(X,\mathscr{B})$ is standard $\kappa$-Borel, by definition it is
$\kappa$-Borel isomorphic to a $\kappa$-Borel subset
$B\subseteq\tensor[^{\kappa}]{\kappa}{}$. Given $A\in\mathscr{B}$, the
subspace $(A,\mathscr{B}\restriction A)$ is then $\kappa$-Borel isomorphic to
a set in
$\mathsf{Bor}_{\kappa}(B)\subseteq\mathsf{Bor}_{\kappa}(\tensor[^{\kappa}]{\kappa}{})$,
hence $(A,\mathscr{B}\restriction A)$ is standard $\kappa$-Borel.

Conversely, assume that $(A,\mathscr{B}\restriction A)$ is standard
$\kappa$-Borel. Let $\tau$ be a $\kappa$-additive $f\mathrm{SC}_{\kappa}$
topology on $X$ with $\mathscr{B}=\mathsf{Bor}_{\kappa}(X,\tau)$, as given by
Theorem 4.5, and use Proposition 4.8 to refine the topology $\tau\restriction
A$ on $A$ (which obviously generates $\mathscr{B}\restriction A$) to a
$\kappa$-additive $f\mathrm{SC}_{\kappa}$ topology $\tau_{A}$ on $A$ such that
$\mathscr{B}\restriction A=\mathsf{Bor}_{\kappa}(A,\tau_{A})$. Let
$\mathcal{B}$ be a clopen basis for $(A,\tau_{A})$ of size $\leq\kappa$. Since
$\mathcal{B}\subseteq\mathsf{Bor}_{\kappa}(A,\tau_{A})=\mathsf{Bor}_{\kappa}(X,\tau)\restriction
A$, for every $U\in\mathcal{B}$ we can find
$C_{U}\in\mathsf{Bor}_{\kappa}(X,\tau)$ such that $U=C_{U}\cap A$. Without
loss of generality, we may assume that the family $\mathcal{C}=\\{C_{U}\mid
U\in\mathcal{B}\\}$ is closed under complements. Define $\hat{\tau}$ to be the
smallest $\kappa$-additive topology on $X$ containing $\tau\cup\mathcal{C}$.
Then $\hat{\tau}$ is Hausdorff because it refines $\tau$, it is zero-
dimensional (and hence regular) because $\tau$ is zero-dimensional and
$\mathcal{C}$ is closed under complements, and it has weight at most
$(\kappa+|\mathcal{C}|)^{<\kappa}=\kappa^{<\kappa}=\kappa$. Hence, by Theorem
2.12 we have that the space $(X,\hat{\tau})$ is $\mathbb{G}$-metrizable.
Furthermore,
$\mathsf{Bor}_{\kappa}(X,\hat{\tau})=\mathsf{Bor}_{\kappa}(X,\tau)$ because
$\tau\subseteq\hat{\tau}\subseteq\mathsf{Bor}_{\kappa}(X,\tau)$, and $A$ is a
$\mathbb{G}$-Polish subspace of $(X,\hat{\tau})$ because by construction
$\hat{\tau}\restriction A=\tau_{A}$ and $\tau_{A}$ is a $\kappa$-additive
$f\mathrm{SC}_{\kappa}$ topology. Therefore, by Corollary 2.26 we have that
$A$ is a $G^{\kappa}_{\delta}$ subset of $(X,\hat{\tau})$, so in particular
$A\in\mathsf{Bor}_{\kappa}(X,\hat{\tau})=\mathsf{Bor}_{\kappa}(X,\tau)=\mathscr{B}$.
∎

######  Corollary 4.10.

Let $X,Y$ be standard $\kappa$-Borel spaces. If $A\subseteq X$ is
$\kappa$-Borel and $f\colon A\to Y$ is a $\kappa$-Borel embedding, then $f(A)$
is $\kappa$-Borel in $Y$.

Corollary 4.10 is the analogue of the classical fact that an injective Borel
image of a Borel set is still Borel (see [Kec95, Section 15.A]). Notice
however that in the generalized version the hypothesis on $f$ is stronger: we
need it to be a $\kappa$-Borel embedding, and not just an injective
$\kappa$-Borel map. This is mainly due to the fact that in the generalized
context we lack the analogue of Luzin’s separation theorem. Indeed, one can
even prove [LS15, Corollary 1.9] that there are non-$\kappa$-Borel sets which
are continuous injective images of the whole $\tensor[^{\kappa}]{\kappa}{}$,
hence our stronger requirement cannot be dropped.

We finally come to the problem of characterizing which topologies induce a
standard $\kappa$-Borel structure. Of course this class is larger than the
collection of all $f\mathrm{SC}_{\kappa}$ topologies, even when restricting to
the $\kappa$-additive case. Indeed, the relative topology on any
$\kappa$-Borel non-$G^{\kappa}_{\delta}$ subspace
$B\subseteq\tensor[^{\kappa}]{\kappa}{}$ generates a standard $\kappa$-Borel
structure, yet it is not $f\mathrm{SC}_{\kappa}$ itself because of Theorems
2.21 and 2.27. On the other hand, if a space $(X,\tau)$ is homeomorphic to a
$\kappa$-Borel subset of $\tensor[^{\kappa}]{\kappa}{}$, then it clearly
generates a standard $\kappa$-Borel structure by definition. Theorems 2.12 and
4.9 allow us to reverse the implication, yielding the desired characterization
in the case of $\kappa$-additive topologies. (For the nontrivial direction,
use the fact that by Theorem 2.12 every $\kappa$-additive space of weight
$\leq\kappa$ is, up to homeomorphism, a subspace of
$\tensor[^{\kappa}]{\kappa}{}$.)

######  Corollary 4.11.

Let $(X,\tau)$ be a $\kappa$-additive space of weight $\leq\kappa$. Then
$(X,\mathsf{Bor}_{\kappa}(X,\tau))$ is a standard $\kappa$-Borel space if and
only if $(X,\tau)$ is _homeomorphic_ to a $\kappa$-Borel subset of
$\tensor[^{\kappa}]{\kappa}{}$ (or, equivalently, of
$\tensor[^{\kappa}]{2}{}$).

In [MR13, Definition 3.6], the author considered _topological_ spaces
$(X,\tau)$ with weight $\leq\kappa$ such that the induced $\kappa$-Borel
structure is $\kappa$-Borel isomorphic to a $\kappa$-Borel subset of
$\tensor[^{\kappa}]{\kappa}{}$. By Corollary 4.11 it turns out that when
$\tau$ is regular Hausdorff and $\kappa$-additive, a space $(X,\tau)$
satisfies [MR13, Definition 3.6] if and only if it is homeomorphic (and not
just $\kappa$-Borel isomorphic) to a $\kappa$-Borel subset of
$\tensor[^{\kappa}]{\kappa}{}$.

##  5\. Final remarks and open questions

In the classical setup, Polish spaces are closed under countable sums,
countable products, and $G_{\delta}$ subspaces. Moving to the generalized
context, all classes considered so far are trivially closed under sums of size
$\leq\kappa$. However, by Theorem 2.31 the class of
$\mathrm{SC}_{\kappa}$-spaces is already lacking closure with respect to
closed subspaces (even when restricting the attention to $\kappa$-additive
spaces or, equivalently, to spherically complete $\mathbb{G}$-Polish spaces).
In view of Proposition 2.10, the class of $f\mathrm{SC}_{\kappa}$-spaces is a
more promising option. Indeed, since such class is also straightforwardly
closed under $\leq\kappa$-products, where the product is naturally endowed by
the $<\kappa$-supported product topology, we easily get:

######  Theorem 5.1.

The class of $f\mathrm{SC}_{\kappa}$-spaces is closed under
$G^{\kappa}_{\delta}$ subspaces and $\leq\kappa$-sized sums and products.

Moving to $\mathbb{G}$-Polish spaces, by Theorem 2.27 we still have closure
under $G^{\kappa}_{\delta}$ subspaces. However, it is then not transparent how
to achieve closure under $\leq\kappa$-sized products. The naïve attempt of
mimicking what is done in the classical case would require to first develop a
theory of convergent $\kappa$-indexed series in some suitable group
$\mathbb{G}$, and then use it to try to define the complete
$\mathbb{G}$-metric on the product. Theorem 2.21 provides an elegant bypass to
these difficulties and directly leads us the the following theorem.

######  Theorem 5.2.

The class of $\mathbb{G}$-Polish spaces (equivalently: $\kappa$-additive
$f\mathrm{SC}_{\kappa}$-spaces) is closed under
$G^{\kappa}_{\delta}$-subspaces and $\leq\kappa$-sized sums and products.

###### Proof.

For $\leq\kappa$-sized products, just notice that both the property of being
$\kappa$-additive and the property of being strongly fair $\kappa$-Choquet are
straightforwardly preserved by such operation. ∎

Moreover, we also get the analogue of Sierpiński’s theorem [Kec95, Theorem
8.19]: the classes of $\mathbb{G}$-Polish spaces and
$f\mathrm{SC}_{\kappa}$-spaces are both closed under continuous open images.
(Notice that a similar result holds for $\mathrm{SC}_{\kappa}$-spaces, as
observed in [CS16, Proposition 2.7].)

######  Theorem 5.3.

Let $X$ be $\mathbb{G}$-Polish, and $Y$ be a space of weight $\leq\kappa$. If
there is a continuous open surjection $f$ from $X$ onto $Y$, then $Y$ is
$\mathbb{G}$-Polish.

The same is true is we replace $\mathbb{G}$-Polishness by the (weaker)
property of being an $f\mathrm{SC}_{\kappa}$-space.

###### Proof.

By Theorem 2.21, it is enough to show that the properties of being strongly
fair $\kappa$-Choquet and being $\kappa$-additive are preserved by $f$. The
former is straightforward. For the latter, let $(U_{\alpha})_{\alpha<\nu}$ be
a sequence of open subsets of $Y$, for some $\nu<\kappa$. If
$\bigcap_{\alpha<\nu}U_{\alpha}\neq\emptyset$, let $y$ be arbitrary in
$\bigcap_{\alpha<\nu}U_{\alpha}$ and, using surjectivity of $f$, let $x\in X$
be such that $f(x)=y$. Since $x\in\bigcap_{\alpha<\nu}f^{-1}(U_{\alpha})$ and
the latter set is open by $\kappa$-additivity of $X$, there is $V\subseteq X$
open such that $x\in V\subseteq\bigcap_{\alpha<\nu}f^{-1}(U_{\alpha})$. It
follows that $f(V)$ is an open neighborhood of $y$ such that
$f(V)\subseteq\bigcap_{\alpha<\nu}U_{\alpha}$, as desired. ∎

There is still one interesting open question related to
$f\mathrm{SC}_{\kappa}$-subspaces of a given space of weight $\leq\kappa$. By
Corollary 2.26, if $X$ is also $\kappa$-additive and $Y\subseteq X$ is an
$f\mathrm{SC}_{\kappa}$-subspace of it, then $Y$ is $G^{\kappa}_{\delta}$ in
$X$. We do not know if the same remains true if we drop $\kappa$-additivity.
The following corollary is the best result we have in this direction: it
follows from Theorem 2.12 and the fact that by $\kappa^{<\kappa}=\kappa$ and
the proof of Proposition 2.33, every (regular Hausdorff) topology of weight
$\leq\kappa$ can be naturally refined to a $\kappa$-additive one in such a way
that the new open sets are $F^{\kappa}_{\sigma}$ (i.e. a $\leq\kappa$-sized
union of closed sets or, equivalently, the complement of a
$G^{\kappa}_{\delta}$ set) in the old topology.

######  Corollary 5.4.

Let $X$ be a space of weight $\leq\kappa$, and $Y\subseteq X$ be an
$f\mathrm{SC}_{\kappa}$-subspace of it. Then $Y$ is a $\leq\kappa$-sized
intersection of $F^{\kappa}_{\sigma}$ subsets of $X$.

It is then natural to ask whether the above computation can be improved.

######  Question 5.5.

In the same hypotheses of Corollary 5.4, is $Y$ a $G_{\delta}^{\kappa}$ subset
of $X$? What if we assume that $X$ be $f\mathrm{SC}_{\kappa}$?

In the literature on generalized descriptive set theory, the notion of an
analytic set is usually generalized as follows.

######  Definition 5.6.

A subset of a space121212Since $f\mathrm{SC}_{\kappa}$-spaces have been
introduced in the present paper, the definition of $\kappa$-analytic sets
given in the literature is of course usually restricted to the spaces
$\tensor[^{\kappa}]{\kappa}{}$ and $\tensor[^{\kappa}]{2}{}$ and their powers.
The only exception is [MR13], where it is given for all $\leq\kappa$-weighted
topologies generating a standard $\kappa$-Borel structure (see [MR13,
Definitions 3.6 and 3.8]). of weight $\leq\kappa$ is $\kappa$-analytic if and
only if it is a continuous image of a closed subset of
$\tensor[^{\kappa}]{\kappa}{}$. A set is $\kappa$-coanalytic if its complement
is $\kappa$-analytic, and it is $\kappa$-bianalytic if it is both
$\kappa$-analytic and $\kappa$-coanalytic.

Although the definition works for a larger class of spaces, in this paper we
will concentrate on subsets of $f\mathrm{SC}_{\kappa}$-spaces. Analogously to
what happens in the classical case, one can then prove that Definition 5.6 is
equivalent to several other variants: for example, a set
$A\subseteq\tensor[^{\kappa}]{\kappa}{}$ is $\kappa$-analytic if and only if
it is the projection of a closed
$C\subseteq(\tensor[^{\kappa}]{\kappa}{})^{2}$, if and only if131313This
reformulation involves only $\kappa$-Borel sets and functions, thus the notion
of a $\kappa$-analytic set is independent on the actual topology. This allows
us to naturally extend this concept to subsets of arbitrary (standard)
$\kappa$-Borel spaces. it is a $\kappa$-Borel image of some set
$B\in\mathsf{Bor}_{\kappa}(\tensor[^{\kappa}]{\kappa}{})$ (see [AMR, Corollary
7.3] and [MR13, Proposition 3.11]). As explained in [LS15, Theorem 1.5], a
major difference from the classical setup is instead that we cannot add among
the equivalent reformulations of $\kappa$-analyticity that of being a
continuous image of the whole $\tensor[^{\kappa}]{\kappa}{}$—this condition
defines a properly smaller class when $\kappa$ is uncountable (and, as usual,
$\kappa^{<\kappa}=\kappa$).

The reason for using Definition 5.6 instead of directly generalizing [Kec95,
Definition 14.1] is precisely that we were still lacking an appropriate notion
of generalized Polish-like space. We can now fill this gap.

######  Proposition 5.7.

Let $X$ be an $f\mathrm{SC}_{\kappa}$-space. For any $A\subseteq X$, the
following are equivalent:

  1. (a)

$A$ is $\kappa$-analytic (i.e. a continuous image of a closed subset of
$\tensor[^{\kappa}]{\kappa}{}$);

  2. (b)

$A$ is a continuous image of a $\mathbb{G}$-Polish space;

  3. (c)

$A$ is a continuous image of an $f\mathrm{SC}_{\kappa}$-space.

###### Proof.

The implications (a) $\Rightarrow$ (b) and (b) $\Rightarrow$ (c) follow from
Theorem 2.21. For the remaining implication (c) $\Rightarrow$ (a), suppose
that $Y$ is an $f\mathrm{SC}_{\kappa}$-space and that $g\colon Y\to X$ is
continuous and onto $A$. Use Proposition 2.33 to refine the topology $\tau$ of
$Y$ to a topology $\tau^{\prime}$ such that $(Y,\tau^{\prime})$ is
$\kappa$-additive and still $f\mathrm{SC}_{\kappa}$. Use Theorem 2.21 again to
find a closed set $C\subseteq\tensor[^{\kappa}]{\kappa}{}$ and a homeomorphism
$f\colon C\to(Y,\tau^{\prime})$: then $g\circ f$ is a continuous surjection
from $C$ onto $A$. ∎

Clearly, in Proposition 5.7(c) we can equivalently consider $\kappa$-additive
$f\mathrm{SC}_{\kappa}$-spaces. We instead cannot restrict ourselves to
$\mathrm{SC}_{\kappa}$-spaces, even when further requiring
$\kappa$-additivity. Indeed, by Theorem 2.31 and [LS15, Proposition 1.3] every
such space is a continuous image of the whole $\tensor[^{\kappa}]{\kappa}{}$:
it follows that the collection of all continuous images of $\kappa$-additive
$\mathrm{SC}_{\kappa}$-spaces coincides with the collection of continuous
images of $\tensor[^{\kappa}]{\kappa}{}$, and it is thus strictly smaller than
the class of all $\kappa$-analytic sets by the mentioned [LS15, Theorem 1.5].

A variant of Definition 5.6 considered in [LS15] is the class
$I^{\kappa}_{\mathrm{cl}}$ of continuous _injective_ images of closed subsets
of $\tensor[^{\kappa}]{\kappa}{}$ (clearly, all such sets are in particular
$\kappa$-analytic). When $\kappa=\omega$ the class $I^{\kappa}_{\mathrm{cl}}$
coincides with Borel sets, but when $\kappa>\omega$ the class
$I^{\kappa}_{\mathrm{cl}}$ is strictly larger than
$\mathsf{Bor}_{\kappa}(\tensor[^{\kappa}]{\kappa}{})$ by [LS15, Corollary
1.9]. Moreover, if $\mathrm{V}=\mathrm{L}[x]$ with $x\subseteq\kappa$, then by
[LS15, Corollary 1.14] all $\kappa$-analytic subsets of
$\tensor[^{\kappa}]{\kappa}{}$ belong to $I^{\kappa}_{\mathrm{cl}}$. This
result can be extended to $\kappa$-analytic subsets of arbitrary
$f\mathrm{SC}_{\kappa}$-spaces.

######  Corollary 5.8.

Assume that $\mathrm{V}=\mathrm{L}[x]$ with $x\subseteq\kappa$, and let $X$ be
an arbitrary $f\mathrm{SC}_{\kappa}$-space. Then every $\kappa$-analytic
$A\subseteq X$ is a continuous injective image of a closed subset of
$\tensor[^{\kappa}]{\kappa}{}$.

###### Proof.

By Corollary 2.34 there is a closed $C\subseteq\tensor[^{\kappa}]{\kappa}{}$
and a continuous bijection $f\colon C\to X$. Notice that $f^{-1}(A)$ is
$\kappa$-analytic in $C$ because the class of $\kappa$-analytic sets is easily
seen to be closed under continuous preimages, hence it is $\kappa$-analytic in
$\tensor[^{\kappa}]{\kappa}{}$ as well. By [LS15, Corollary 1.14] there is a
continuous injection from some closed $D\subseteq\tensor[^{\kappa}]{\kappa}{}$
onto $f^{-1}(A)$, which composed with $f$ gives the desired result. ∎

We are now going to show that the class $I^{\kappa}_{\mathrm{cl}}$ can be
characterized through changes of topology.

######  Theorem 5.9.

Let $(X,\tau)$ be an $f\mathrm{SC}_{\kappa}$-space and $A\subseteq X$. Then
the following are equivalent:

  1. (a)

$A\in I^{\kappa}_{\mathrm{cl}}$;

  2. (b)

there is an $f\mathrm{SC}_{\kappa}$ topology $\tau^{\prime}$ on $A$ such that
$\tau^{\prime}\supseteq\tau\restriction A$.

###### Proof.

Suppose first that $C\subseteq\tensor[^{\kappa}]{\kappa}{}$ is closed and
$f\colon C\to X$ is a continuous injection with range $A$. Let $\tau^{\prime}$
be obtained by pushing forward along $f$ the (relative) topology of $C$, so
that $(A,\tau^{\prime})$ and $C$ are homeomorphic. Then $(A,\tau^{\prime})$ is
an $f\mathrm{SC}_{\kappa}$-space by Theorem 2.21, and $\tau^{\prime}$ refines
$\tau\restriction A$ because $f$ was continuous.

Conversely, if $(A,\tau^{\prime})$ is an $f\mathrm{SC}_{\kappa}$-space then by
Theorem 2.21 again there is a closed $C\subseteq\tensor[^{\kappa}]{\kappa}{}$
and a homeomorphism $f\colon C\to(A,\tau^{\prime})$. Since
$\tau^{\prime}\supseteq\tau\restriction A$, it follows that $C$ and $f$
witness $A\in I^{\kappa}_{\mathrm{cl}}$. ∎

This also allows us to precisely determine to what extent the technique of
change of topology discussed in Section 4 can be applied to non-$\kappa$-Borel
sets.

######  Corollary 5.10.

Let $(X,\tau)$ be an $f\mathrm{SC}_{\kappa}$-space.

  1. (1)

Let $A\subseteq X$. If there is an $f\mathrm{SC}_{\kappa}$ topology
$\tau^{\prime}\supseteq\tau$ on $X$ such that $A$ is $\tau^{\prime}$-clopen
(or even just $A\in\mathsf{Bor}_{\kappa}(X,\tau^{\prime})$), then $A$ is
$\kappa$-bianalytic.

  2. (2)

If $\mathrm{V}=\mathrm{L}[x]$ with $x\subseteq\kappa$, then for all
$\kappa$-bianalytic $A\subseteq X$ there is a $\kappa$-additive
$f\mathrm{SC}_{\kappa}$ topology $\tau^{\prime}\supseteq\tau$ on $X$ such that
$A$ is $\tau^{\prime}$-clopen.

###### Proof.

For part (1) observe that since $A$ is $\tau^{\prime}$-clopen, then by
Proposition 2.10 both $A$ and $X\setminus A$ are
$f\mathrm{SC}_{\kappa}$-spaces when endowed with the relativization of
$\tau^{\prime}$. Therefore by Theorem 5.9 they are in
$I^{\kappa}_{\mathrm{cl}}$, and thus $\kappa$-analytic. If instead of $A$
being $\tau^{\prime}$-clopen we just have
$A\in\mathsf{Bor}_{\kappa}(X,\tau^{\prime})$, use Proposition 4.1 to further
refine $\tau^{\prime}$ to a suitable $\tau^{\prime\prime}$ turning $A$ into a
$\tau^{\prime\prime}$-clopen set, and then apply the previous argument to
$\tau^{\prime\prime}$ instead of $\tau^{\prime}$.

We now move to part (2). By Corollary 5.8, under our assumption all
$\kappa$-analytic subsets of $X$ are in $I^{\kappa}_{\mathrm{cl}}$. It follows
that for every $\kappa$-bianalytic set $B\subseteq X$ there is a continuous
bijection $f\colon C\to X$ with $C\subseteq\tensor[^{\kappa}]{\kappa}{}$
closed and $f^{-1}(B)$ clopen relatively to $C$: just fix $f_{0}\colon
C_{0}\to B$ and $f_{1}\colon C_{1}\to X\setminus B$ witnessing $B\in
I^{\kappa}_{\mathrm{cl}}$ and $X\setminus B\in I^{\kappa}_{\mathrm{cl}}$,
respectively, let $C$ be the sum of $C_{0}$ and $C_{1}$, and set $f=f_{0}\cup
f_{1}$. Pushing forward the topology of $C$ along $f$ we then get the desired
$\tau^{\prime}$ (the fact that $\tau^{\prime}\supseteq\tau_{b}$ follows again
from the continuity of $f$). ∎

Corollary 5.10 justifies our claim that there might be non-$\kappa$-Borel sets
that can be turned into clopen sets via a nice change of topology (see item
(b) on page (b)). Indeed, when $\kappa$ is uncountable there are
$\kappa$-bianalytic subsets of $\tensor[^{\kappa}]{\kappa}{}$ which are not
$\kappa$-Borel (see e.g. [FHK14, Theorem 18]), and Corollary 5.10(2) applies
to them.

Having extended the notion of a $\kappa$-analytic set to arbitrary
$f\mathrm{SC}_{\kappa}$-spaces, it is natural to ask whether the deep analysis
carried out in [LS15] can be transferred to such wider context. Some of the
results have already been explicitly extended in this paper, see e.g.
Corollaries 2.15, 4.2, and 5.8, which extend, respectively, [LS15, Proposition
1.3, Lemma 1.11, and Corollary 1.14]. Other results naturally transfer to our
general setup using the ideas developed so far.

######  Question 5.11.

Which other results from [LS15] hold for $\kappa$-analytic subsets of
arbitrary $f\mathrm{SC}_{\kappa}$-spaces? For example, for which
$f\mathrm{SC}_{\kappa}$-spaces $X$ is there a closed $C\subseteq X$ which is
not a continuous image of the whole $\tensor[^{\kappa}]{\kappa}{}$, or a
non-$\kappa$-Borel set $A\subseteq X$ which is an injective continuous image
of $\tensor[^{\kappa}]{\kappa}{}$?

Similar questions can be raised about the analogue of the Hurewicz dichotomy
for $\kappa$-analytic subsets of $\tensor[^{\kappa}]{\kappa}{}$ studied in
[LMRS16].

We now move to generalizations of the perfect set property.

######  Definition 5.12.

Let $X$ be a space. A set $A$ has the $\kappa$-perfect set property
($\kappa$-PSP for short) if either $|A|\leq\kappa$ or $A$ contains a closed
set homeomorphic to $\tensor[^{\kappa}]{2}{}$.

The $\kappa$-Borel version of the $\kappa$-PSP would then read as follows:
either $|A|\leq\kappa$ or $A$ contains a $\kappa$-Borel set which is
$\kappa$-Borel isomorphic to $\tensor[^{\kappa}]{2}{}$. However, for most
applications it is convenient to consider a slightly stronger reformulation.

######  Definition 5.13.

Let $X$ be a space. A set $A$ has the Borel $\kappa$-perfect set property
($\mathsf{Bor}_{\kappa}$-PSP for short) if either $|A|\leq\kappa$ or there is
a _continuous_ $\kappa$-Borel embedding $f\colon\tensor[^{\kappa}]{2}{}\to A$
with $f(\tensor[^{\kappa}]{2}{})\in\mathsf{Bor}_{\kappa}(X)$.

By Corollary 4.10, if the $\kappa$-Borel structure of $X$ is standard then the
fact that $f(\tensor[^{\kappa}]{2}{})\in\mathsf{Bor}_{\kappa}(X)$ follows from
the other conditions. Notice also that the $\mathsf{Bor}_{\kappa}$-PSP is in
general strictly weaker than the $\kappa$-PSP. For example, consider the space
$X=\tensor[^{\kappa}]{2}{}$ equipped with the _product_ topology. It is a
$\kappa$-perfect $\mathrm{SC}_{\kappa}$-space, hence $X$ itself and all its
open subsets have the $\mathsf{Bor}_{\kappa}$-PSP (see Corollary 5.14 below).
However, $X$ is compact: thus its clopen subsets cannot contain a closed
homeomorphic copy of the generalized Cantor space, which clearly is not
compact, and thus they do not have the $\kappa$-PSP.

In Definitions 5.12 and 5.13 we are of course allowing the special case $A=X$.
With this terminology, Theorem 3.6 asserts that the $\kappa$-PSP holds for all
$\kappa$-additive $\kappa$-perfect $\mathrm{SC}_{\kappa}$-spaces. From this
and Proposition 2.33, we can easily infer the following fact, which is just a
more precise formulation of [CS16, Proposition 3.1]. (Of course here we are
also using that if $\tau$ is $\kappa$-perfect, then the topology from the
proof of Proposition 2.33 is still $\kappa$-perfect.)

######  Corollary 5.14.

If $X$ is a nonempty $\kappa$-perfect $\mathrm{SC}_{\kappa}$-space, then there
is a continuous $\kappa$-Borel embedding from $\tensor[^{\kappa}]{2}{}$ into
$X$ (with a $\kappa$-Borel range, necessarily). In particular, the
_$\mathsf{Bor}_{\kappa}$ -PSP_ holds for $\kappa$-perfect
$\mathrm{SC}_{\kappa}$-spaces.

It is instead independent of ${\sf ZFC}$ whether the (Borel) $\kappa$-perfect
set property holds for ($\kappa$-additive) $f\mathrm{SC}_{\kappa}$-spaces.
Indeed, if there is a $\kappa$-Kurepa tree $T$ with $<2^{\kappa}$-many
branches, then no $\kappa$-PSP-like property can hold for $[T]$ because of
cardinality reasons. On the other hand, in [Sch17] the third author
constructed a model of ${\sf ZFC}$ where all “definable” subsets of
$\tensor[^{\kappa}]{\kappa}{}$ (including e.g. all $\kappa$-analytic sets and
way more) have the $\mathsf{Bor}_{\kappa}$-PSP: combining Proposition 2.33
with Theorem 2.21 we then get that such property holds for arbitrary
$f\mathrm{SC}_{\kappa}$-spaces and their “definable” subsets. Indeed, the same
reasoning combined with Proposition 4.8 can be used to show that _if_ the
$\mathsf{Bor}_{\kappa}$-PSP holds for all closed subsets of, say,
$\tensor[^{\kappa}]{\kappa}{}$, then it automatically propagates to all
$\kappa$-Borel subsets of all $f\mathrm{SC}_{\kappa}$-spaces. Moreover, we can
even just start from superclosed sets (equivalently, up to homeomorphism, from
$\kappa$-additive $\mathrm{SC}_{\kappa}$-spaces). Indeed, if
$C=[T]\subseteq\tensor[^{\kappa}]{\kappa}{}$ is closed, then arguing as in the
proof of Lemma 2.36 we can construct a superclosed set
$C^{\prime}=[T^{\prime}]$ such that $C\subseteq C^{\prime}$,
$|C^{\prime}|\leq\max\\{|C|,\kappa\\}$, and all points in $C^{\prime}\setminus
C$ are isolated in $C^{\prime}$. It follows that if the
$\mathsf{Bor}_{\kappa}$-PSP holds for $C^{\prime}$ then it holds also for $C$
because if $f\colon\tensor[^{\kappa}]{2}{}\to C^{\prime}$ is a continuous
injection then $f(\tensor[^{\kappa}]{2}{})\subseteq C$ (use the fact that
$\tensor[^{\kappa}]{2}{}$ is perfect). Summing up we thus have:

######  Theorem 5.15.

The following are equivalent:

  1. (a)

the _$\mathsf{Bor}_{\kappa}$ -PSP_ holds for superclosed subsets of
$\tensor[^{\kappa}]{\kappa}{}$;

  2. (b)

the _$\mathsf{Bor}_{\kappa}$ -PSP_ holds for closed subsets of
$\tensor[^{\kappa}]{\kappa}{}$;

  3. (c)

the _$\mathsf{Bor}_{\kappa}$ -PSP_ holds for all ($\kappa$-additive)
$f\mathrm{SC}_{\kappa}$-spaces;

  4. (d)

the _$\mathsf{Bor}_{\kappa}$ -PSP_ holds for all $\kappa$-Borel subsets of all
$f\mathrm{SC}_{\kappa}$-spaces.

The Borel $\kappa$-perfect set property for $f\mathrm{SC}_{\kappa}$-spaces has
important consequences for their classification up to $\kappa$-Borel
isomorphisms.

######  Corollary 5.16.

Suppose that the _$\mathsf{Bor}_{\kappa}$ -PSP_ holds for (super)closed
subsets of $\tensor[^{\kappa}]{\kappa}{}$. If $X$ is an
$f\mathrm{SC}_{\kappa}$-space with $|X|>\kappa$, then $X$ is $\kappa$-Borel
isomorphic to $\tensor[^{\kappa}]{2}{}$. In particular, any two
$f\mathrm{SC}_{\kappa}$-spaces $X,Y$ are $\kappa$-Borel isomorphic if and only
if $|X|=|Y|$.

In particular, if the _$\mathsf{Bor}_{\kappa}$ -PSP_ holds for (super)closed
subsets of $\tensor[^{\kappa}]{\kappa}{}$ then up to $\kappa$-Borel
isomorphism the generalized Cantor space $\tensor[^{\kappa}]{2}{}$ is the
unique $f\mathrm{SC}_{\kappa}$-space of size $>\kappa$.

###### Proof.

By our assumption and Theorem 5.15, $\tensor[^{\kappa}]{2}{}$ is
$\kappa$-Borel isomorphic to a $\kappa$-Borel subsets of $X$. Conversely, $X$
is $\kappa$-Borel isomorphic to a $\kappa$-Borel subset of
$\tensor[^{\kappa}]{2}{}$ by Theorem 4.5 and the fact that
$\tensor[^{\kappa}]{2}{}$ and $\tensor[^{\kappa}]{\kappa}{}$ are
$\kappa$-Borel isomorphic. Thus the result follows from the natural
$\kappa$-Borel version of the usual Cantor-Schröder-Bernstein argument. ∎

Using the same argument and Corollary 5.14 we also get that when restricting
to $\kappa$-perfect $\mathrm{SC}_{\kappa}$-spaces the conclusions of Corollary
5.16 hold unconditionally—see [CS16, Corollary 3.7].

When dealing with topological game theory, one often wonders about what kind
of winning strategies the players have at disposal in the given game. In this
context, one can differentiate between perfect information strategies, that
need to know all previous moves in order to be able to give an answer, and
tactics, that instead rely only on the last move to determine the answer. The
two notions do not coincide in general: there are games where a player has a
winning strategy, but not a winning tactic. For example, [Deb85] describes a
topological space where player $\mathrm{II}$ has a winning strategy but no
winning tactic in the classical strong Choquet game (see also [Lup14]). Debs’
example can easily be adapted to show that there exists a
(non-$\kappa$-additive) topological space of weight $\kappa$ where player
$\mathrm{II}$ has a winning strategy but not a winning tactic in
$f\mathrm{G}^{s}_{\kappa}(X)$ (or in $\mathrm{G}^{s}_{\kappa}(X)$), or that
there is a $\kappa$-additive topological space of weight $>\kappa$ with the
same property. In contrast, Proposition 2.13 implies that for
$\kappa$-additive spaces of weight $\leq\kappa$ the two notions of winning
tactic and winning strategy can be used interchangeably.

######  Corollary 5.17.

Let $X$ be a $\kappa$-additive space of weight $\leq\kappa$. Then
$\mathrm{II}$ has a winning strategy in $f\mathrm{G}^{s}_{\kappa}(X)$ (resp.
$\mathrm{G}^{s}_{\kappa}(X)$) if and only if she has a winning tactic.

###### Proof.

For the nontrivial direction, by Proposition 2.13 we can restrict the
attention to (super)closed subsets of $\tensor[^{\kappa}]{\kappa}{}$, so let
$X=[T]$ for some pruned tree $T\subseteq\tensor[^{<\kappa}]{\kappa}{}$. Then
any function $\sigma\colon\tau\to\tau$ that associate to every nonempty open
set $U\subseteq[T]$ a basic open set $\mathbf{N}_{s}\cap[T]\subseteq U$ for
some $s\in T$ is a winning tactic for $\mathrm{II}$ in
$f\mathrm{G}^{s}_{\kappa}([T])$. Indeed, the answers $N_{s_{\alpha}}\cap[T]$
of $\sigma$ at every round $\alpha$ are such that $s_{\alpha}\subseteq
s_{\beta}$ for any $\alpha<\beta<\kappa$. Hence, if the game does not stop
before $\kappa$-many rounds, then the final intersection
$\bigcap_{\alpha<\kappa}\mathbf{N}_{s_{\alpha}}\cap[T]$ is nonempty, since it
contains $s=\bigcup_{\alpha<\kappa}s_{\alpha}$ (or any sequence extendeding
$s$, if $s$ has length $<\kappa$). A similar argument shows that if $T$ is
superclosed, than the tactic described above is winning also for
$\mathrm{G}^{s}_{\kappa}([T])$. ∎

For more details about perfect information strategies and tactics, and for
some interesting problems in the field, see for example [Sch07].

In this paper we generalized metrics by allowing values in structures
different from $\mathbb{R}$. Another possible generalization of metric spaces
is given by uniform spaces. In this context we have a notion of completeness
as well, which is however strictly weaker than the notions we considered so
far. Indeed, all $\mathbb{G}$-metrizable spaces of weight $\leq\kappa$ (that
is, by Theorem 2.12, all subspaces of $\tensor[^{\kappa}]{\kappa}{}$) are
paracompact and Hausdorff, and this entails that they are completely
uniformizable. It follows that any non-$G^{\kappa}_{\delta}$ subset of
$\tensor[^{\kappa}]{\kappa}{}$ is a completely uniformizable space of weight
$\leq\kappa$ which is not $f\mathrm{SC}_{\kappa}$ and, more generally, that
the class of completely uniformizable spaces of weight $\leq\kappa$ properly
extends the class of all $\kappa$-additive spaces with weight $\leq\kappa$
(recall that we are tacitly restricting to regular Hausdorff spaces). Thus by
Theorem 4.9 such class contains spaces which are not even $\kappa$-Borel
isomorphic to an $f\mathrm{SC}_{\kappa}$-space (that is, they are not standard
$\kappa$-Borel): this seems to rule out the possibility of developing a decent
(generalized) descriptive set theory in such a generality. Nevertheless, from
the topological perspective it would still be interesting to know whether this
property also extends the class of non-$\kappa$-additive
$f\mathrm{SC}_{\kappa}$-spaces or at least $\mathrm{SC}_{\kappa}$-spaces.

######  Question 5.18.

Is every $f\mathrm{SC}_{\kappa}$-space completely uniformizable?

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