2.1. Topologies, Borel structures and $\sigma $ -topologies

In this paper, we will be dealing extensively with different topologies and Borel structures. We therefore begin by carefully fixing some basic conventions.

For a topological space X, we will denote its topology, that is, lattice of open sets by

(2.1.1) $$ \begin{align} \mathcal{O}(X) \subseteq \mathcal{P}(X). \end{align} $$

We will occasionally need to consider multiple topologies on the same underlying set, which will be denoted $\mathcal {S}, \mathcal {T}, \mathcal {T}', \dotsc \subseteq \mathcal {P}(X)$ ; we reserve the notation $\mathcal {O}(X)$ for a distinguished topology that X is considered to be ‘equipped’ with, for which we are willing to abuse notation as usual and denote the topological space by X instead of $(X,\mathcal {O}(X))$ .

By a Borel space, we will mean what is commonly called a measurable space, that is, a set equipped with an arbitrary $\sigma $ -algebra of subsets (not assumed to be induced by any topology), which are called Borel. Similarly to (2.1.1), we will denote the Borel $\sigma $ -algebra of a Borel space X by

(2.1.2) $$ \begin{align} \mathcal{B}(X) \subseteq \mathcal{P}(X). \end{align} $$

For a topological space X, we equip it by default with the $\sigma $ -algebra generated by $\mathcal {O}(X)$ , as usual. A standard Borel space is one whose $\sigma $ -algebra is generated by a Polish topology.

We will be working with nonmetrizable spaces, for which we use the modified Borel hierarchy due to Selivanov [Reference Selivanov31]. The main difference from the usual definition is in level 2:

(2.1.3)

for countably many open $U_i, V_i \in \mathcal {O}(X) =: \boldsymbol {\Sigma }^0_1(X)$ , where $(U_i \Rightarrow V_i) := \{x \in X \mid x \in U_i \implies x \in V_i\}$ . For higher countable ordinals $\xi> 2$ , we may define $\boldsymbol {\Sigma }^0_\xi $ in the same way but taking $U_i, V_i \in \boldsymbol {\Sigma }^0_{\zeta _i}(X)$ for $\zeta _i < \xi $ (and $\boldsymbol {\Pi }^0_\xi $ to be the complements of $\boldsymbol {\Sigma }^0_\xi $ sets), but here, it is enough to take $U_i = X$ as in the usual definition for metrizable spaces. As usual, $\boldsymbol {\Delta }^0_\xi := \boldsymbol {\Sigma }^0_\xi \cap \boldsymbol {\Pi }^0_\xi $ .

The following simple facts take the place of the $T_1$ or Hausdorff axioms in many arguments in the non-Hausdorff context and will be freely used without mention:

1. (2.1.4) Points in $T_0$ first-countable spaces are $\boldsymbol {\Pi }^0_2$ .

2. (2.1.5) The equality relation in a $T_0$ second-countable space X is $\boldsymbol {\Pi }^0_2$ in $X^2$ .

The following common generalization of topologies and $\sigma $ -algebras will allow us to unify some analogous statements between the topological versus Borel contexts:

Definition 2.1.7. If $\star $ is a binary operation on sets, then for two families of sets $\mathcal {S}$ and $\mathcal {T}$ , we write

(Typically $\mathcal {S}, \mathcal {T}$ are closed under countable unions, over which $\star $ distributes.)

For instance, if $\mathcal {S} \subseteq \mathcal {P}(X)$ and $\mathcal {T} \subseteq \mathcal {P}(Y)$ are $\sigma $ -topologies, then $\mathcal {S} \otimes \mathcal {T} \subseteq \mathcal {P}(X \times Y)$ is the product $\sigma $ -topology, consisting of all countable unions of rectangles. Thus, if $\mathcal {S}, \mathcal {T}$ are second-countable topologies, then $\mathcal {S} \otimes \mathcal {T}$ is the product topology. See also Definitions 3.2.1 and 4.2.1.

We now recall some basic notions and facts surrounding Baire category. In a topological space X, a subset $A \subseteq X$ is comeager if it contains a countable intersection of dense open sets and meager if its complement $\neg A$ is comeager. We write

(2.1.8)

(2.1.9)

We say X is a Baire space if the Baire category theorem holds in it, that is, every comeager set is dense, or equivalently every nonempty open set is nonmeager, and X is completely Baire if every closed subspace $Y \subseteq X$ is Baire, or equivalently every $\boldsymbol {\Pi }^0_2$ subspace is Baire (see, e.g., [Reference Chen4, 7.2]). For $A, B \subseteq X$ , we have the relations of containment and equality mod meager:

(2.1.10)

(2.1.11)

We say that $A \subseteq X$ has the Baire property if it is $=^*$ to an open set; such sets include all open sets and form a $\sigma $ -algebra, hence include all Borel sets. Explicitly, we have the following formulas which show how to inductively construct, for each Borel set $A \subseteq X$ , an open set $U_A =^* A$ :

(2.1.12) $$ \begin{align} U_{\bigcup_i A_i} &:= \bigcup_i U_{A_i},\qquad\ \ \,\kern1pt\qquad\qquad\qquad\qquad\qquad\qquad \end{align} $$

(2.1.13) $$ \begin{align} U_{A \setminus B} &:= U_A \cap (\neg U_B)^\circ = \bigcup_{W \in \mathcal{W}; W \cap U_B = \varnothing} (W \cap U_A), \end{align} $$

where $\mathcal {W}$ is any open basis for X (and $(-)^\circ $ denotes interior). These formulas will be particularly useful for working uniformly with ‘bundles’ of spaces; see Section 2.3.

2.2. Quasi-Polish spaces

The main topological setting of this paper is de Brecht’s quasi-Polish spaces [Reference de Brecht10] (see also [Reference Chen4]), which can be defined via (2.2.8) or (2.2.9) below. We list here some basic properties of quasi-Polish spaces we will freely use; for proofs, see the aforementioned references.

1. (2.2.1) All quasi-Polish spaces are $T_0$ , second-countable, and completely Baire.

2. (2.2.2) A topological space is Polish iff it is quasi-Polish and regular ( $T_3$ ).

3. (2.2.3) The Sierpinski space $\mathbb {S} = \{0, 1\}$ , where $\{1\}$ is open but not closed, is quasi-Polish.

4. (2.2.4) If X is a quasi-Polish space and $A_i \in \boldsymbol {\Sigma }^0_\xi (X)$ are countably many sets, then there is a finer quasi-Polish topology containing each $A_i$ and contained in $\boldsymbol {\Sigma }^0_\xi (X)$ . In more detail,

   1. (a) adjoining a single $\boldsymbol {\Delta }^0_2$ set to the topology of X preserves quasi-Polishness;

   2. (b) if the intersection of countably many quasi-Polish topologies contains a quasi-Polish topology, then their union generates a quasi-Polish topology.

5. (2.2.5) A quasi-Polish space can be made zero-dimensional Polish by adjoining countably many closed sets to the topology, hence is in particular standard Borel (in the usual sense).

6. (2.2.6) Countable products of quasi-Polish spaces are quasi-Polish.

7. (2.2.7) A space with a countable cover by open quasi-Polish subspaces is quasi-Polish.

8. (2.2.8) A subspace of a quasi-Polish space is quasi-Polish iff it is $\boldsymbol {\Pi }^0_2$ (in the sense of (2.1.3)). In fact, quasi-Polish spaces are precisely the $\boldsymbol {\Pi }^0_2$ subspaces of $\mathbb {S}^{\mathbb {N}}$ , up to homeomorphism.

9. (2.2.9) A continuous open $T_0$ quotient of a quasi-Polish space is quasi-Polish. In fact, nonempty quasi-Polish spaces are precisely the continuous open $T_0$ quotients of $\mathbb {N}^{\mathbb {N}}$ .

This last property ultimately underlies all of the topological realization results in this paper.

2.3. Fiberwise topology

Given any function $f : X \to Y$ between sets, we may regard X as a bundle over Y, that is, as a family of sets, the fibers $f^{-1}(y)$ , indexed over $y \in Y$ . In general, when we refer to an f -fiberwise concept or property, we will mean that it occurs simultaneously on each fiber.

Definition 2.3.2. Recall that for any functions $f : X \to Y$ and $g : Z \to Y$ , we have the pullback or fiber product

$$ \begin{align*} Z \times_Y X := \{(z,x) \in Z \times X \mid g(z) = f(x)\}, \end{align*} $$

which fits into a commutative square

(2.3.3)

where $\pi _1, \pi _2$ are the projections, such that each $\pi _1$ -fiber $\pi _1^{-1}(z)$ is in canonical bijection (via $\pi _2$ ) with the f-fiber $f^{-1}(g(z))$ . In this situation, we also call $\pi _1$ the pullback of f along g.

If X has an f-fiberwise topology, we may therefore transfer it via $\pi _2$ to a pullback $\pi _1$ -fiberwise topology on $Z \times _Y X$ . Henceforth, whenever we have a pullback of a bundle of topological spaces, we will by default regard it as being equipped with the pullback fiberwise topology.

The following trivial facts, as well as their various analogs (e.g., (2.3.7) below), will play a key role. For a pullback square as above, the Beck–Chevalley condition says that for $A \subseteq X$ ,

(2.3.4) $$ \begin{align} g^{-1}(f(A)) = \pi_1(\pi_2^{-1}(A)). \end{align} $$

A special case is Frobenius reciprocity: For $Z \subseteq Y$ and g the inclusion,

(2.3.5) $$ \begin{align} Z \cap f(A) = f(f^{-1}(Z) \cap A). \end{align} $$

Definition 2.3.6. Let $f : X \to Y$ be a bundle of topological spaces. For any $A \subseteq X$ , we define the Baire category quantifiers

$$ \begin{align*} \exists^*_f(A) &:= \{y \in Y \mid \exists^* x \in f^{-1}(y)\, (x \in A)\}, \\ \forall^*_f(A) &:= \{y \in Y \mid \forall^* x \in f^{-1}(y)\, (x \in A)\} = \neg \exists^*_f(\neg A) \end{align*} $$

(where the $\exists ^*, \forall ^*$ are with respect to the fiberwise topology on $f^{-1}(y)$ ). For $A, B \subseteq X$ , we write

Note that these reduce to the ‘absolute’ Baire category notions (2.1.8)–(2.1.11) when $Y = 1$ .

We now record some basic facts about fiberwise Baire category, most of which are usually stated in the special case of a product bundle $X = Y \times Z$ (and $f = \pi _1$ ) but generalize straightforwardly; see [Reference Kechris23, §8.J], [Reference Chen4, §7]. First, from the fact that $\exists ^*_f$ is defined fiberwise, we clearly have:

1. (2.3.7) (Beck–Chevalley condition) For a pullback square as in (2.3.3),

   $$ \begin{align*} g^{-1}(\exists^*_f(A)) = \exists^*_{\pi_1}(\pi_2^{-1}(A)). \end{align*} $$

2. (2.3.8) (Frobenius reciprocity) In particular, for $Z \subseteq Y$ ,

   $$ \begin{align*} Z \cap \exists^*_f(A) = \exists^*_f(f^{-1}(Z) \cap A). \end{align*} $$

3. (2.3.9) In particular, for $Z \subseteq \exists ^*_f(X)$ ,

   $$ \begin{align*} Z = \exists^*_f(f^{-1}(Z)). \end{align*} $$

The next few properties are direct fiberwise translations of basic properties of ‘global’ Baire category from Section 2.1:

1. (2.3.10) $\exists ^*_f(A) \subseteq f(A)$ , with equality if A is fiberwise open and X is fiberwise Baire.

2. (2.3.11) Thus, if X is fiberwise Baire and $A =^*_f U \in \mathcal {O}_f(X)$ , then $\exists ^*_f(A) = \exists ^*_f(U) = f(U)$ .

3. (2.3.12) $\exists ^*_f$ preserves countable unions; $\forall ^*_f$ preserves countable intersections.

The following are fiberwise translations of the formulas (2.1.12) and (2.1.13):

1. (2.3.13) For countably many $A_i \subseteq X$ , if each $A_i =^*_f U_{A_i}$ , then $\bigcup _i A_i =^*_f U_{\bigcup _i A_i} := \bigcup _i U_{A_i}$ .

2. (2.3.14) If $A =^*_f U_A \in \mathcal {O}_f(X)$ and $B =^*_f U_B \in \mathcal {O}_f(X)$ , then for any f-fiberwise basis $\mathcal {W} \subseteq \mathcal {O}_f(X)$ ,

   $$ \begin{align*} A \setminus B =^*_f U_{A \setminus B} := \bigcup_{W \in \mathcal{W}} (W \cap U_A \setminus f^{-1}(f(W \cap U_B))). \end{align*} $$

Applying $\exists ^*_f$ to this last formula yields, assuming X is fiberwise Baire and for a countable fiberwise basis $\mathcal {W}$ , using (2.3.11) and (2.3.5),

(2.3.15) $$ \begin{align} \exists^*_f(A \setminus B) &= \bigcup_{W \in \mathcal{W}} (\exists^*_f(W \cap A) \setminus \exists^*_f(W \cap B)). \end{align} $$

By induction on $\xi $ , these last few properties yield the following; see [Reference Kechris23, 22.22], [Reference Chen4, 7.5].

2.4. Borel fiberwise topology

We now consider bundles of spaces $f : X \to Y$ , where the base space Y is standard Borel, and the fiberwise topology is ‘uniformly Borel’.

The key tool enabling a well-behaved theory of such ‘Borel fiberwise topology’ is the following classical result. It is usually stated for the case of a product bundle $X = Y \times Z$ , with $\mathcal {S}$ consisting of the cylinders $Y \times U$ for U in some countable open basis for Z; see [Reference Kechris23, 28.7]. However, essentially the same proof yields the general form below, which was pointed out in [Reference Chen6, 8.14].

Definition 2.4.2. Let $f : X \to Y$ be a Borel map between standard Borel spaces, and suppose X is equipped with an f-fiberwise topology. We call X a standard Borel bundle of quasi-Polish spaces over Y if X is fiberwise quasi-Polish, and is ‘uniformly fiberwise second-countable’ in that it has a countable fiberwise open (sub)basis consisting of Borel sets $\mathcal {U} \subseteq \mathcal {B}(X)$ . For such X, we let

$$ \begin{align*} \mathcal{BO}_f(X) := \mathcal{B}(X) \cap \mathcal{O}_f(X) \end{align*} $$

denote the $\sigma $ -topology of Borel f-fiberwise open sets in X.

It is perhaps not obvious that this is the ‘correct’ definition of a ‘uniformly quasi-Polish’ bundle of spaces. The definition of quasi-Polish space requires not only that the topology is ‘countably generated’ (i.e., second-countable), but also ‘countably presented’ (i.e., $\boldsymbol {\Pi }^0_2$ ; see (2.2.8)); why do we require uniformity of only the former but not the latter? The following shows that it is automatic (recall the notion of a compatible $\sigma $ -topology from Definition 2.2.10):

Proof. (a) is an immediate consequence of Kunugui–Novikov (Theorem 2.4.1).

(b) Let $\mathcal {U} \subseteq \mathcal {BO}_f(X)$ be a countable Borel fiberwise open basis for X, including each given $U_i$ . First, suppose that each $U \in \mathcal {U}$ is in fact fiberwise clopen. We then have a fiberwise embedding

$$ \begin{align*} e : X &\longrightarrow Y \times 2^{\mathcal{U}} \\ x &\longmapsto (f(x), (\chi_U(x))_{U \in \mathcal{U}}), \end{align*} $$

where each $\chi _U$ is the characteristic function of U; this is a fiberwise embedding because $\mathcal {U}$ is a fiberwise basis. By the Lusin–Suslin theorem (see, e.g., [Reference Kechris23, 15.1]), $e(X) \subseteq Y \times 2^{\mathcal {U}}$ is Borel. Since e is a fiberwise embedding, its image is ( $\pi _1$ -)fiberwise $G_\delta $ ; thus, by Saint Raymond’s uniformization theorem for Borel fiberwise $G_\delta $ sets [Reference Kechris23, 35.45], together with Kunugui–Novikov,

$$ \begin{align*} e(X) = \bigcap_i \bigcup_j (B_{ij} \times V_{ij}) \end{align*} $$

for some countably many Borel $B_{ij} \subseteq Y$ and open $V_{ij} \subseteq 2^{\mathcal {U}}$ . Find any compatible zero-dimensional Polish topology on Y making these $B_{ij}$ clopen. Then $e(X) \subseteq Y \times 2^{\mathcal {U}}$ is $G_\delta $ , hence we may pull back the subspace topology along e to X, yielding a compatible zero-dimensional Polish topology.

If each $U \in \mathcal {U}$ is merely fiberwise open, we may run the above argument replacing the role of $2^{\mathcal {U}}$ with $\mathbb {S}^{\mathcal {U}}$ , using the following quasi-Polish generalization of Saint Raymond’s theorem, to get a compatible quasi-Polish topology on X, homeomorphic to a $\boldsymbol {\Pi }^0_2$ subspace of $Y \times \mathbb {S}^{\mathcal {U}}$ .

Lemma 2.4.4. Let Y be a standard Borel space, Z be a quasi-Polish space, $A \subseteq Y \times Z$ be a $\pi _1$ -fiberwise $\boldsymbol {\Pi }^0_2$ set. Then there are countably many Borel $B_{ij} \subseteq Y$ and open $U_i, V_{ij} \subseteq Z$ such that

Proof. By (2.2.9), let $Z'$ be Polish and $g : Z' \twoheadrightarrow Z$ be a continuous open surjection. Then ${A' := (Y \times g)^{-1}(A) \subseteq Y \times Z'}$ is $\pi _1$ -fiberwise $G_\delta $ , hence by Saint Raymond’s theorem

$$ \begin{align*} A' = \bigcap_{i'} \bigcup_j (B_{i'j} \times V^{\prime}_{i'j}) \end{align*} $$

for countably many Borel $B_{i'j} \subseteq Y$ and open $V^{\prime }_{i'j} \subseteq Z'$ . Then

(with the $(Y \times g)$ -fiberwise open basis $Y \times \mathcal {U}$ for $Y \times Z$ , where $\mathcal {U}$ is any open basis for Z)

This is clearly of the desired form, where i runs over all pairs $(i',U)$ .

Among standard Borel bundles of quasi-Polish spaces $f : X \to Y$ , the best behaved are those for which ‘fiberwise nonemptiness of a Borel fiberwise open $U \subseteq X$ ’ can be detected in a uniformly Borel way. These bundles may be characterized as follows:

We call the bundle Borel-overt if these equivalent conditions hold (borrowing a term from constructive topology; see, for example, [Reference Spitters34]).

Borel-overt bundles are the ones for which ‘fiberwise Baire category is Borel’:

Finally, in this subsection, we recall the Kuratowski–Ulam theorem, which has the following conceptual formulation in terms of bundles:

Theorem 2.4.8 (Kuratowski–Ulam).

Let $X \xrightarrow {f} Y \xrightarrow {g} Z$ be Borel maps between Borel spaces. Suppose that X is equipped with a $(g \circ f)$ -fiberwise quasi-Polish topology, and Y is equipped with a g-fiberwise quasi-Polish topology such that both topologies are fiberwise compatible with the subspace Borel structures on each fiber, f is fiberwise continuous and fiberwise open over Z and $\exists ^*_f, \exists ^*_g$ preserve Borel sets. (For example, $f, g$ could both be continuous open maps between quasi-Polish spaces, or more generally g could be a standard Borel-overt bundle of such.)

Then

$$ \begin{align*} \exists^*_g \circ \exists^*_f = \exists^*_{g \circ f} : \mathcal{B}(X) \longrightarrow \mathcal{B}(Z). \end{align*} $$

Since $\exists ^*$ is defined fiberwise, it is equivalent to just consider the case where $Z = 1$ is a singleton, where the statement becomes: For a continuous open $f : X \to Y$ between quasi-Polish spaces, a Borel set $A \subseteq X$ is nonmeager in X iff for nonmeagerly many $y \in Y$ , $f^{-1}(y) \cap A$ is nonmeager in $f^{-1}(y)$ . The classical case is when f is a product projection; see [Reference Kechris23, 8.41]. It was pointed out in [Reference Melleray and Tsankov26, A.1] that essentially the same proof works for a continuous open f between Polish spaces and in [Reference Chen4, 7.6] that the quasi-Polish case works just as well. See also Theorem 5.2.16 below.

2.5. Lower powerspaces

Definition 2.5.1. For a topological space X, its lower powerspace $\mathcal {F}(X)$ is the space of closed subsets of X, equipped with the lower Vietoris topology generated by the subbasic open sets

$$ \begin{align*} \Diamond U := \{F \in \mathcal{F}(X) \mid F \cap U \ne \varnothing\} \quad \text{for }U \in \mathcal{O}(X). \end{align*} $$

We record some elementary properties:

1. (2.5.2) $\Diamond : \mathcal {O}(X) \to \mathcal {O}(\mathcal {F}(X))$ preserves unions; thus, restricting to a basis for $\mathcal {O}(X)$ still yields a subbasis for $\mathcal {O}(\mathcal {F}(X))$ . In particular, if X is second-countable, then so is $\mathcal {F}(X)$ .

2. (2.5.3) If X is $T_0$ , we have a continuous embedding ${\downarrow } : X \to \mathcal {F}(X)$ , $x \mapsto \overline {\{x\}}$ (where $\overline {(-)}$ denotes closure), with ${\downarrow }^{-1}(\Diamond U) = U$ .

3. (2.5.4) A continuous map $f : X \to Y$ induces the continuous image-closure map $\overline {f} : \mathcal {F}(X) \to \mathcal {F}(Y)$ , $F \mapsto \overline {f(F)}$ , with $\overline {f}^{-1}(\Diamond V) = \Diamond f^{-1}(V)$ . Thus, if f is an embedding, then so is $\overline {f}$ .

4. (2.5.5) For two spaces $X, Y$ , the Cartesian product map $\times : \mathcal {F}(X) \times \mathcal {F}(Y) \to \mathcal {F}(X \times Y)$ is continuous, with $\times ^{-1}(\Diamond (U \times V)) = \Diamond U \times \Diamond V$ .

We will also need the following generalization:

Definition 2.5.7. For a continuous map $f : X \to Y$ , regarded as a bundle of topological spaces, its fiberwise lower powerspace $\mathcal {F}_Y(X) = \mathcal {F}_f(X)$ is the space of pairs $(y,F)$ , where $y \in Y$ and $F \in \mathcal {F}(f^{-1}(y))$ , regarded as a bundle via the first projection $\pi _1 : \mathcal {F}_Y(X) \to Y$ , and equipped with the topology generated by the subbasic open sets

$$ \begin{align*} \Diamond_V U :={}& \{(y,F) \in \mathcal{F}_Y(X) \mid y \in V \ {\text{and}}\ F \cap U \ne \varnothing\} \\ ={}& \pi_1^{-1}(V) \cap \Diamond_Y U \quad \text{for }U \in \mathcal{O}(X)\text{ and }V \in \mathcal{O}(Y). \end{align*} $$

In other words, $\mathcal {F}_Y(X)$ is equipped with the topology induced by the embedding

(2.5.8)

By abuse of notation, we will often refer to an element of $\mathcal {F}_Y(X)$ in the fiber over a fixed $y \in Y$ as just a closed set $F \in \mathcal {F}(f^{-1}(y))$ in that fiber, rather than the pair $(y,F)$ .

Proposition 2.5.9 [Reference Chen6, 2.2].

If X above is completely Baire while Y is $T_0$ first-countable, then the image of the above embedding is

$$ \begin{align*} \bigcap_{U \in \mathcal{O}(X), V \in \mathcal{O}(Y)} ((V \times \Diamond U) \Leftrightarrow (Y \times \Diamond(f^{-1}(V) \cap U))), \end{align*} $$

where the intersection may be taken over any bases of $\mathcal {O}(X), \mathcal {O}(Y)$ . Thus, if moreover $X, Y$ are second-countable, then the image is $\boldsymbol {\Pi }^0_2$ ; and if $X, Y$ are quasi-Polish, then so is $\mathcal {F}_Y(X)$ .

Analogously to (2.5.2)–(2.5.5), we have

1. (2.5.10) $(V, U) \mapsto \Diamond _V U$ preserves unions in both variables (and binary intersections in V).

2. (2.5.11) If X is f-fiberwise $T_0$ over Y, we have a continuous embedding ${\downarrow }_Y := (f, {\downarrow }) : X \to \mathcal {F}_Y(X)$ over Y, with ${\downarrow }^{-1}(\Diamond _V U) = f^{-1}(V) \cap U$ .

3. (2.5.12) For continuous maps $X \xrightarrow {f} Y \xrightarrow {g} Z$ , we get a continuous map $\overline {f} : \mathcal {F}_Z(X) \to \mathcal {F}_Z(Y)$ over Z, with $\overline {f}^{-1}(\Diamond _W V) = \Diamond _W f^{-1}(V)$ .

4. (2.5.13) For continuous maps $f : X \to Z$ and $g : Y \to Z$ , the fiber product map $\times _Z : \mathcal {F}_Z(X) \times _Z \mathcal {F}_Z(Y) \to \mathcal {F}_Z(X \times _Z Y)$ is continuous, with $\times ^{-1}(\Diamond _W (U \times _Z V)) = \Diamond _W U \times _Z \Diamond _W V$ .

2.6. Linear quantifiers

In this paper, our main interest in (fiberwise) lower powerspaces stems from the conceptual link they provide between ‘quantifier-like maps’ on open and Borel sets, such as the Baire category quantifier $\exists ^*_f$ , and topological realization. We now make this precise. These ideas are more-or-less well known in the point-free topology literature, for which see [Reference Joyal and Tierney22], [Reference Vickers37] and Section 5.3. To keep this paper accessible, we give here a classical point-based treatment.

Proposition 2.6.2. For any topological spaces $X, Y$ , we have a canonical bijection

Definition 2.6.3. Let $f : X \to Z$ and $g : Y \to Z$ be continuous maps, regarded as bundles of spaces. An $\mathcal {O}(Z)$ -linear map $\phi : \mathcal {O}(X) \to \mathcal {O}(Y)$ is a linear map which moreover obeys

$$ \begin{align*} \phi(f^{-1}(W) \cap U) = g^{-1}(W) \cap \phi(U) \quad \forall W \in \mathcal{O}(Z),\, U \in \mathcal{O}(X). \end{align*} $$

We are particularly interested in the case where $g = 1_Y$ is an identity, where this becomes

$$ \begin{align*} \phi(f^{-1}(V) \cap U) = V \cap \phi(U) \quad \forall V \in \mathcal{O}(Y),\, U \in \mathcal{O}(X); \end{align*} $$

in this case, we also call $\phi $ a $\mathcal {O}(Y)$ -linear quantifier. If moreover

$$ \begin{align*} \phi(f^{-1}(V)) = V \quad \text{or equivalently} \quad \phi(X) = Y \end{align*} $$

(cf. (2.3.9)), we call $\phi $ a $\mathcal {O}(Y)$ -linear retraction of $f^{-1} : \mathcal {O}(Y) \to \mathcal {O}(X)$ .

Similarly, if $f : X \to Y$ is a Borel map between Borel spaces, we have the notion of a $\mathcal {B}(Y)$ -linear quantifier or $\mathcal {B}(Y)$ -linear retraction $\phi : \mathcal {B}(X) \to \mathcal {B}(Y)$ , defined via the same equations.

Proposition 2.6.6. For continuous maps $f : X \to Z$ and $g : Y \to Z$ , where X is completely Baire and Z is $T_0$ first-countable, the bijection of Proposition 2.6.2 induces a bijection

Thus, in particular, for $f : X \to Y = Z$ and $g = 1_Y$ , we have

with the $\mathcal {O}(Y)$ -linear retractions of $f^{-1}$ corresponding to the continuous sections of $\pi _1$ picking a nonempty closed set from each fiber of f.

Proof. A continuous map $h : Y \to \mathcal {F}_Z(X)$ is the same thing as a continuous map $(h_1, h_2) : Y \to Z \times \mathcal {F}(X)$ which lands in the image of the embedding (2.5.8); to say that h is ‘over Z’ means $h_1 = \pi _1 \circ h = g$ . So the right-hand side of the first bijection equivalently consists of continuous maps $h_2 : Y \to \mathcal {F}(X)$ such that $(g, h_2) : Y \to Z \times \mathcal {F}(X)$ lands in the image of (2.5.8). By Proposition 2.5.9, this happens iff for each $U \in \mathcal {O}(X)$ and $W \in \mathcal {O}(Z)$ , we have

$$ \begin{align*} g^{-1}(W) \cap h_2^{-1}(\Diamond U) = (g,h_2)^{-1}(W \times \Diamond U) = (g,h_2)^{-1}(Z \times \Diamond(f^{-1}(W) \cap U)) = h_2^{-1}(\Diamond(f^{-1}(W) \cap U)); \end{align*} $$

by Proposition 2.6.2, such $h_2$ are in bijection with $\mathcal {O}(Z)$ -linear maps $\phi : \mathcal {O}(X) \to \mathcal {O}(Y)$ .

If $g = 1_Y$ , to say that h always picks a nonempty set is to say that $h_2$ does, that is, $h_2^{-1}(\Diamond X) = Y$ , which by Proposition 2.6.2 corresponds to $\phi (X) = Y$ .

We now give yet a third description of linear quantifiers:

Definition 2.6.7. Let $f : X \to Y$ be a continuous map, $\phi : \mathcal {O}(X) \to \mathcal {O}(Y)$ be an $\mathcal {O}(Y)$ -linear quantifier which corresponds via Proposition 2.6.6 to a continuous section $h : Y \to \mathcal {F}_Y(X)$ of $\pi _1 : \mathcal {F}_Y(X) \to Y$ . The support of $\phi $ is

$$ \begin{align*} \mathrm{supp}(\phi) :={}& \{x \in X \mid x \in h(f(x))\} \\ ={}& \{x \in X \mid \forall U \in \mathcal{O}(X)\, (x \in U \implies f(x) \in \phi(U))\} \\ ={}& \bigcap_{U \in \mathcal{O}(X)} (U \Rightarrow f^{-1}(\phi(U))). \end{align*} $$

Clearly, this is an f-fiberwise closed subset of X, which is $\boldsymbol {\Pi }^0_2$ if X is second-countable (since it suffices to intersect over basic open U).

Proposition 2.6.9. For a continuous map $f : X \to Y$ , where X is completely Baire and Y is $T_0$ first-countable, we have a bijection

with the $\mathcal {O}(Y)$ -linear retractions of $f^{-1}$ corresponding to the F such that $f(F) = Y$ .

Using this correspondence, we may extend (2.2.9) to certain nonopen quotient maps:

Remark 2.6.12. The Borel versions of linear quantifiers from Definition 2.6.3 correspond to a standard descriptive set-theoretic notion. Let $f : X \to Y$ be a Borel map between standard Borel spaces. For a $\mathcal {B}(Y)$ -linear quantifier $\phi : \mathcal {B}(X) \to \mathcal {B}(Y)$ , we may chase through the proof of Proposition 2.6.2 to get a map $h : Y \to \{\text {linear maps } \mathcal {B}(X) \to \mathbb {S}\}$ ; now, such a linear map is the same thing as a $\sigma $ -ideal in $\mathcal {B}(X)$ so that $\phi $ corresponds to a family of $\sigma $ -ideals $(\mathcal {I}_y)_{y \in Y}$ , from which $\phi $ is recovered via

$$ \begin{align*} \phi(A) = \{y \in Y \mid h(y)(A) = 1\} = \{y \in Y \mid A \not\in \mathcal{I}_y\}. \end{align*} $$

In lieu of ‘continuity’ of h as in Proposition 2.6.2, we have the requirement that $A \in \mathcal {B}(X) \implies \phi (A) \in \mathcal {B}(Y)$ , which is a weak (nonparametrized) form of the requirement that $(\mathcal {I}_y)_y$ is a Borel on Borel family; see [Reference Kechris23, 18.5]. And $\mathcal {B}(X)$ -linearity means in particular that for each $y \in Y$ ,

which means that each $\mathcal {I}_y$ is determined by its restriction to $\mathcal {B}(f^{-1}(y))$ . So we have a bijection

$$ \begin{align*} \{\mathcal{B}(Y)\text{-linear quant. } \mathcal{B}(X) \to \mathcal{B}(Y)\} &\cong \{\text{weakly Borel on Borel fam. of }\sigma\text{-ideals }\mathcal{I}_y \subseteq \mathcal{B}(f^{-1}(y))\}. \end{align*} $$

The $\mathcal {B}(Y)$ -linear retractions of $f^{-1}$ correspond to the families of proper $\sigma $ -ideals $\mathcal {I}_y \subsetneq \mathcal {B}(f^{-1}(y))$ .

Note that if X is equipped with a (nice) topology, then a linear quantifier on Borel sets may be restricted to one on open sets. Using this observation, we have

Proof. Note first that $\mathcal {O}(Y)$ is indeed a topology: It is closed under countable unions because $\phi $ preserves countable unions, hence closed under arbitrary unions by second-countability of $\mathcal {O}(X)$ ; it contains $Y = \phi (X)$ ; and it is closed under binary intersections because by $\mathcal {B}(Y)$ -linearity,

(2.6.14) $$ \begin{align} \phi(U) \cap \phi(V) = \phi(f^{-1}(\phi(U)) \cap V) \end{align} $$

for $U, V \in \mathcal {O}(X)$ , and $f^{-1}(\phi (U)) \in f^{-1}(\phi (\mathcal {O}(X))) \subseteq \mathcal {O}(X)$ by assumption. By this same assumption, with this topology on Y, f is continuous, and Y is $T_0$ since $\mathcal {O}(Y)$ separates points. Now, apply Theorem 2.6.10.

2.7. Baire category for coarser topologies

For our main topological realization results below, we will be applying Corollary 2.6.13 above to $\mathcal {B}(Y)$ -linear quantifiers $\phi $ given by the Baire category quantifier $\exists ^*_f$ with respect to some f-fiberwise topology on X which is coarser than the restriction of the global topology on X. We now specialize the machinery of the preceding subsection to this case.

Lemma 2.7.2. Let X be a set with two topologies $\mathcal {S} \subseteq \mathcal {T}$ , and let $Y \subseteq X$ be the $\mathcal {T}$ -support of $\mathcal {S}$ . Suppose that every $\mathcal {T}$ -open set has the $\mathcal {S}$ -Baire property. Then the inclusion $(Y,\mathcal {T}|Y) \to (X,\mathcal {S})$ induces an isomorphism of Baire category algebras

$$ \begin{align*} \mathcal{B}(Y,\mathcal{T}|Y)/\text{meager} \cong \mathcal{B}(X,\mathcal{S})/\text{meager}. \end{align*} $$

In particular, an $\mathcal {S}$ -Borel set is $\mathcal {S}$ -(co)meager iff its restriction to Y is $\mathcal {T}$ -(co)meager, and $(Y,\mathcal {T}|Y)$ is a Baire space.

Proof. We claim that a $\mathcal {T}$ -closed $F \subseteq Y$ is $\mathcal {T}|Y$ -nowhere dense iff it is $\mathcal {S}$ -meager. Indeed, if F is $\mathcal {S}$ -meager, then so is $Y \Rightarrow F$ since $\neg Y$ is $\mathcal {S}$ -meager by definition of support, whence the $\mathcal {T}$ -interior of $Y \Rightarrow F$ is disjoint from Y again by definition of support, which means the $\mathcal {T}|Y$ -interior of F is empty, that is, F is $\mathcal {T}|Y$ -nowhere dense. Conversely, if F is $\mathcal {T}|Y$ -nowhere dense, then letting $U \subseteq X$ be $\mathcal {S}$ -open such that $F \triangle U$ is $\mathcal {S}$ -meager, we have that $U \setminus F$ is $\mathcal {T}$ -open and $\mathcal {S}$ -meager, hence disjoint from Y by definition of support, that is, $Y \cap U \subseteq F$ , whence $Y \cap U = \varnothing $ since F is $\mathcal {T}|Y$ -nowhere dense, whence $F = F \setminus U$ is $\mathcal {S}$ -meager.

It follows that for $\mathcal {S}$ -meager $A \subseteq X$ , A is contained in the union of countably many $\mathcal {S}$ -closed nowhere dense sets, whose intersections with Y are $\mathcal {T}|Y$ -nowhere dense, whence $Y \cap A$ is $\mathcal {T}|Y$ -meager. Thus, the restriction map

$$ \begin{align*} Y \cap (-) : \mathcal{B}(X,\mathcal{S}) \longrightarrow \mathcal{B}(Y,\mathcal{T}|Y) \end{align*} $$

descends to a well-defined map between the category algebras, which is surjective by the assumption that every $\mathcal {T}$ -open set has the $\mathcal {S}$ -Baire property. To check that it is injective: If $A \subseteq X$ such that $Y \cap A$ is $\mathcal {T}|Y$ -meager, then $Y \cap A$ is contained in the union of countably many $\mathcal {T}|Y$ -closed nowhere dense sets, which are $\mathcal {S}$ -meager, whence $Y \cap A$ is $\mathcal {S}$ -meager, whence so is $A \subseteq Y \Rightarrow (Y \cap A)$ again since $\neg Y$ is $\mathcal {S}$ -meager by definition of support. This implies that $(Y,\mathcal {T}|Y)$ is a Baire space because a meager open set must be the restriction of some $\mathcal {S}$ -meager $\mathcal {T}$ -open $U \subseteq X$ , which is disjoint from Y by definition of support.

Applying the preceding lemma fiberwise to this situation, we get

Corollary 2.7.4. In the situation of the preceding remark, suppose furthermore that each $\mathcal {O}(X)$ -open set has the fiberwise $\mathcal {O}_f(X)$ -Baire property. Then for every $\mathcal {O}(X)$ -Borel $A \subseteq X$ , we have

$$ \begin{align*} \exists^*_f(A) = \exists^*_{f|\mathrm{supp}(\exists^*_f)}(\mathrm{supp}(\exists^*_f) \cap A), \end{align*} $$

where $\exists ^*$ on the right-hand side is with respect to the finer topology $\mathcal {O}(X)|\mathrm {supp}(\exists ^*_f)$ , and this finer topology is f-fiberwise Baire.

3.1. Generalities on group actions

Let G be a group acting on a set X. Throughout this section, we always denote the group multiplication by $\mu : G \times G \to G$ , and the action map by $\alpha = \alpha _X : G \times X \to X$ . Note that a special case is the left translation action $\alpha = \mu $ of G on itself.

Definition 3.1.1. As a bundle over X, any action $\alpha $ is isomorphic to the product projection $\pi _2$ , via the following twist involution which we denote by $\dagger $ :

When we apply $\dagger $ to a concept (element, subset, etc.) in $G \times X$ , we say that it twists to the result.

Now, suppose G is a topological group.

Next, consider the associativity axiom $(g \cdot h) \cdot x = g \cdot (h \cdot x)$ .This is expressed by commutativity of the following associativity square, in which we let $\alpha _2$ be the common composite:

(3.1.10)

Just as $\alpha $ is the twisted version of $\pi _2$ so can this entire square be seen as a twisted version of an obviously-commuting square of projections, via the following ‘higher-order twists’:

(3.1.11)

Here, $\pi _{ij}$ is the projection onto the ith and jth coordinates. Note that the right-hand square clearly exhibits $G \times G \times X$ as (an isomorphic copy of) the pullback of $\pi _2$ with itself. Thus, the left-hand associativity square (3.1.10) exhibits $G \times G \times X$ as the pullback of $\alpha $ with itself.

Since $\alpha $ is equipped with a fiberwise topology, the pullback square (3.1.10) yields both a $(\mu \times X)$ -fiberwise topology and a $(G \times \alpha )$ -fiberwise topology on $G \times G \times X$ (recall here Definition 2.3.2). On the other hand, the ‘higher-order twist’ of (3.1.11) also yields an $\alpha _2$ -fiberwise topology on $G \times G \times X$ , corresponding to the $\pi _3$ -fiberwise topology given by the product topology on $G \times G$ .

Proof. The $\alpha _2$ -fiberwise topology restricted to the fibers of $G \times \alpha $ corresponds, via the above diagram (3.1.11), to the $\pi _3$ -fiberwise topology restricted to the fibers of $\pi _{13}$ ; in other words, it is the result of transporting the $\pi _{13}$ -fiberwise topology along (the top edge of) the topmost quadrilateral in (3.1.11). This quadrilateral may be factored as follows:

The right square is a homeomorphism of the $\pi _{13}$ -fiberwise topology on $G \times G \times X$ (it moves the fiber over $(g,x)$ to that over $(g^{-1},gx)$ , followed by the fiberwise homeomorphism $h \mapsto hg^{-1}$ ), which transports along the left triangle to the $(G \times \alpha )$ -fiberwise topology (since the left triangle is $G \times {}$ the triangle in Definition 3.1.1), which is thus equal to the restriction of the $\alpha _2$ -fiberwise topology.

Similarly, the diagram

shows that the $\pi _{23}$ -fiberwise topology transports to both the $(\mu \times X)$ -fiberwise topology and the restriction of the $\alpha _2$ -fiberwise topology.

Now, consider an equivariant map $f : X \to Y$ between two actions of G. Equivariance means commutativity of the left square below, which ‘untwists’ to the right square:

(3.1.13)

Note that when $f = \alpha _X : G \times X \to X$ , where G acts on $G \times X$ via left translation on the first coordinate, the left equivariance square becomes the associativity square (3.1.10), while the entire diagram here is similar but not identical to the diagram (3.1.11) (the difference being that here, we only ‘untwist’ the vertical edges). As before, it is clear that these squares are pullbacks, whence

3.2. Vaught transforms

We now suppose that G is a Polish group and X is a standard Borel G-space. Then the $\alpha $ -fiberwise topology of Definition 3.1.2 turns $G \times X$ into a standard Borel-overt bundle of quasi-Polish spaces over X since $\pi _2$ clearly does. By Remark 3.1.3, a countable Borel fiberwise open basis is given by $\mathcal {U} \times X$ for any countable open basis $\mathcal {U}$ for G.

In the rest of this section, we will use the letters $U, V, W$ to denote arbitrary Borel subsets of G (not necessarily open) and $A, B, C$ to denote arbitrary Borel subsets of X.

Definition 3.2.1. We will use the term Vaught transform to refer loosely to several things. Most generally, it will refer to the Baire category quantifier

$$ \begin{align*} \exists^*_\alpha : \mathcal{B}(G \times X) \longrightarrow \mathcal{B}(X) \end{align*} $$

for the $\alpha $ -fiberwise topology (which lands in $\mathcal {B}(X)$ by Corollary 2.4.7): for Borel $D \subseteq G \times X$ ,

$$ \begin{align*} \exists^*_\alpha(D) = \exists^*_{\pi_2}(D^\dagger) :={}& \{x \in X \mid \exists^* g \in G\, ((g^{-1},gx) \in D)\} \\ ={}& \{x \in X \mid \exists^* g \in G\, ((g,g^{-1}x) \in D)\}. \end{align*} $$

For a Borel rectangle $D = U \times A$ , we use the notation, also called the Vaught transform,

$$ \begin{align*} U * A := \exists^*_\alpha(U \times A) = \{x \in X \mid \exists^* g \in G\, (g \in U \ {\text{and}}\ x \in gA)\}. \end{align*} $$

In the original notation of Vaught (for nonempty open U), this would be denoted $A^{\triangle U^{-1}}$ ; see [Reference Kechris23, 16.2], [Reference Becker and Kechris1, 5.1.7]. However, we find this binary $*$ notation more convenient, to highlight the analogy with the product set under the action $U \cdot A = f(U \times A)$ (see (3.2.2) below).

Following Definition 2.1.7, for $\mathcal {S} \subseteq \mathcal {B}(X)$ (e.g., a compatible quasi-Polish topology), we write Footnote ²

$$ \begin{align*} \mathcal{O}(G) \circledast \mathcal{S} := \exists^*_\alpha(\mathcal{O}(G) \otimes \mathcal{S}) = \{\bigcup_i (U_i * A_i) \mid U_i \in \mathcal{O}(G) \ {\text{and}}\ A_i \in \mathcal{S}\} \subseteq \mathcal{B}(X), \end{align*} $$

where i runs over a countable set. The notation $\mathcal {B}(G) \circledast \mathcal {S}$ has the analogous meaning.

We now record several basic properties of Vaught transforms. These are mostly well known, at least for open U; see [Reference Kechris23, §16.B], [Reference Becker and Kechris1, 5.1.7]. Our main goal here is to make clear that these can all be derived ‘algebraically’ from the corresponding properties of $\exists ^*_\alpha $ from Section 2.3, hence generalize essentially verbatim to the groupoid and point-free contexts in Sections 4.2 and 5.4.

By (2.3.10), Remark 3.1.3, (3.1.5) and (3.1.7),

1. (3.2.2) $U * A \subseteq U \cdot A$ , with equality if U is open and A is orbitwise open.

2. (3.2.3) Thus, if U is nonempty open and A is G-invariant, then $U * A = A$ .

By (2.3.12), $*$ distributes over countable unions:

(3.2.4) $$ \begin{align} (\bigcup_i U_i) * (\bigcup_j A_j) &= \bigcup_{i,j} (U_i * A_j). \end{align} $$

By (2.3.15) applied to $\alpha |(U \times X)$ , for open $U \subseteq G$ and any countable open basis $\mathcal {W}$ for G (so $W \times X$ for $\mathcal {W} \ni W \subseteq U$ form an $\alpha $ -fiberwise open basis for $U \times X$ ),

(3.2.5) $$ \begin{align} U * (A \setminus B) &= \bigcup_{\mathcal{W} \ni W \subseteq U} ((W * A) \setminus (W * B)). \end{align} $$

Thus, for a quasi-Polish G-space X, by induction, or by Proposition 2.3.16,

(3.2.6) $$ \begin{align} \mathcal{O}(G) \circledast \boldsymbol{\Sigma}^0_\xi(X) \subseteq \boldsymbol{\Sigma}^0_\xi(X). \end{align} $$

By Remark 3.1.3 and (3.1.6),

(3.2.7) $$ \begin{align} U \subseteq G \text{ meager} &\implies U * A \subseteq U * X = \varnothing, \nonumber\\ A \subseteq X \text{ orbitwise meager} &\implies U * A \subseteq G * A = \varnothing. \end{align} $$

Since $*$ preserves union 3.2.4, it follows that (recalling notation from Equation (2.1.10), Definition 2.3.6)

(3.2.8) $$ \begin{align} U \subseteq^* V \ {\text{and}}\ A \subseteq^*_G B \implies U * A \subseteq V * B \end{align} $$

(indeed, $U \subseteq ^* V \implies U * A \subseteq ((U \setminus V) \cup V) * A = ((U \setminus V) * A) \cup (V * A) = V * A$ , and similarly for $A \subseteq ^*_G B$ ). We will refer to this law as Pettis’s theorem (for actions); the original Pettis’s theorem for groups follows by taking $\alpha = \mu $ and $U, A \in \mathcal {O}(G)$ .

Note that for any $U \in \mathcal {B}(G)$ , letting $U =^* U' \in \mathcal {O}(G)$ by the Baire property by Equation (3.2.8) we have

(3.2.9) $$ \begin{align} U * A = U' * A. \end{align} $$

Thus, considering $U * A$ for Borel U is really no more general than taking only open U.

Next, consider applying the Baire category quantifier to the various edges of the associativity square (3.1.10). Note that for $U, V \in \mathcal {B}(G)$ and $A \in \mathcal {B}(X)$ , the quantifier $\exists ^*_{G \times \alpha }$ maps $U \times V \times A \mapsto U \times (V * A)$ ; here, we are implicitly using that $G \times G \times X$ is equipped with the $(G \times \alpha )$ -fiberwise topology given by pulling back the $\alpha $ -fiberwise topology as in the square (3.1.10). Similar considerations apply to $\exists ^*_{\mu \times G}$ . Thus, the Beck–Chevalley condition (2.3.7) applied both ways to the pullback square (3.1.10) yields, for $D \in \mathcal {B}(G \times X)$ ,

(3.2.10) $$ \begin{align} \alpha^{-1}(\exists^*_\alpha(D)) &= \exists^*_{G \times \alpha}((\mu \times X)^{-1}(D)) = \exists^*_{\mu \times X}((G \times \alpha)^{-1}(D)), \end{align} $$

which for a rectangle $D = U \times A$ means (using 3.2.4)

(3.2.11) $$ \begin{align} \alpha^{-1}(U * A) &= \exists^*_{G \times \alpha}(\mu^{-1}(U) \times A) = \bigcup_{VW \subseteq U} (V \times (W * A)) \quad \text{for }U, V, W \in \mathcal{O}(G) \end{align} $$

(3.2.12) $$ \begin{align} &= \exists^*_{\mu \times X}(U \times \alpha^{-1}(A)) = U * \alpha^{-1}(A).\qquad \end{align} $$

Note that an immediate consequence of 3.2.11 and (3.1.5) is

1. (3.2.13) For $U \in \mathcal {O}(G)$ and $A \in \mathcal {B}(X)$ , $U * A$ is orbitwise open, that is, $\mathcal {O}(G) \circledast \mathcal {B}(X) \subseteq \mathcal {O}_G(X)$ .

Also, from Lemma 3.1.12, both the $(G \times \alpha )$ - and $(\mu \times X)$ -fiberwise topologies on $G \times G \times X$ are restrictions of the same $\alpha _2$ -fiberwise topology, and both maps are fiberwise open over X to the $\alpha $ -fiberwise topology since this is clearly true of the ‘untwisted’ maps $\pi _{13}, \pi _{23}$ in the right square of (3.1.11). Thus, by the Kuratowski–Ulam Theorem 2.4.8, we have

(3.2.14) $$ \begin{align} \exists^*_\alpha \circ \exists^*_{\mu \times X} = \exists^*_\alpha \circ \exists^*_{G \times \alpha} : \mathcal{B}(G \times G \times X) \to \mathcal{B}(X), \end{align} $$

which on rectangles $U \times V \times A$ says

(3.2.15) $$ \begin{align} (U * V) * A &= U * (V * A). \end{align} $$

For $U, V \subseteq G$ open, we furthermore have

(3.2.16) $$ \begin{align} = (U \cdot V) * A &= U \cdot (V * A) \end{align} $$

by (3.2.2) and (3.2.13).

Finally, for a Borel G-equivariant map $f : X \to Y$ between two standard Borel G-spaces, from the Beck–Chevalley condition (2.3.7) and Lemma 3.1.14, generalizing Equation (3.2.12), we have for $B \in \mathcal {B}(Y)$

(3.2.17) $$ \begin{align} f^{-1}(U * B) = U * f^{-1}(B). \end{align} $$

In fact, this law characterizes equivariance of f; see Corollary 3.4.4 below.

3.3. Topological realization

The following may be regarded as the core result underlying the Becker–Kechris topological realization theorem, generalized to the quasi-Polish context (recall the notation $\circledast $ from Definition 3.2.1):

We now state a consequence of Theorem 3.3.1 that subsumes most commonly used topological realization results as special cases. Recall the notation $\otimes $ for product $\sigma $ -topology from Definition 2.1.7, and the notion of a compatible $\sigma $ -topology from Definition 2.2.10.

The next result is perhaps not usually viewed as a ‘topological realization theorem’. However, we can (somewhat perversely) regard it as such: It says that if the G-action preserves a preexisting topology, then we can find a topological realization ‘compatible with’ that preexisting topology, that is, equal to it. The analogous result for groupoid actions looks less perverse; see Corollary 4.3.7.

It is worth explicitly restating Theorem 3.3.2 in the special case $\mathcal {S} = \mathcal {B}(X)$ , to characterize all Borel sets which are ‘potentially open’ in some topology making the action continuous. For a standard Borel G-space X, recalling the orbitwise topology $\mathcal {O}_G(X)$ from Definition 3.1.4, let

(3.3.7) $$ \begin{align} \mathcal{BO}_G(X) := \mathcal{B}(X) \cap \mathcal{O}_G(X) \end{align} $$

denote the Borel orbitwise open sets. By (3.1.5) and Kunugui–Novikov,

(3.3.8) $$ \begin{align} A \in \mathcal{BO}_G(X) \iff \alpha^{-1}(A) \in \mathcal{O}(G) \otimes \mathcal{B}(X). \end{align} $$

3.4. Equivariant maps

Let G be a Polish group. By [Reference Becker and Kechris1, 2.6.1], $\mathcal {F}(G)^{\mathbb {N}}$ is a universal standard Borel G-space, that is, every other standard Borel G-space admits a Borel equivariant embedding into $\mathcal {F}(G)^{\mathbb {N}}$ . Here, $\mathcal {F}(G)$ is the Effros Borel space of G, that is, the underlying standard Borel space of the lower powerspace of Section 2.5, equipped with the left translation action of G. Since we are working in the quasi-Polish setting, where we have available the lower Vietoris topology on $\mathcal {F}(G)$ , we point out that in fact,

Proof. For any $B \in \mathcal {O}(Y)$ , to say that $f^{-1}(U \cdot B) = U \cdot f^{-1}(B)$ for all $U \in \mathcal {O}(G)$ means $f^{-1}(h_B^{-1}(\Diamond U)) = h_{f^{-1}(B)}^{-1}(\Diamond U)$ for all subbasic $\Diamond U \in \mathcal {O}(\mathcal {F}(G))$ , that is, the triangle

commutes. If this holds for all B, then since $h_{\mathcal {O}(Y)} : Y \to \mathcal {F}(G)^{\mathcal {O}(Y)}$ is an equivariant embedding, and $h_{f^{-1}(\mathcal {O}(Y))}$ is equivariant, we get that f is equivariant.

We also take this opportunity to point out the following universal property enjoyed by the topological realization constructed by Theorem 3.3.1:

Proposition 3.4.6. Let G be a Polish group, X be a quasi-Polish space equipped with a Borel action of G such that $\mathcal {O}(G) \circledast \mathcal {O}(X) \subseteq \mathcal {O}(X)$ . Then any continuous equivariant map $f : X \to Y$ into another quasi-Polish G-space is in fact continuous from the coarser topology $\mathcal {O}(G) \circledast \mathcal {O}(X)$ . In other words, letting $X'$ be X with this coarser topology, the identity map $1_X : X \to X'$ is the universal continuous map from X into a quasi-Polish G-space, hence exhibits $X'$ as the universal continuous ‘completion’ of X:

3.5. Open relations

For a standard Borel G-space X, Corollary 3.3.9 (and more generally Theorem 3.3.2) give precise characterizations of which Borel sets $A \subseteq X$ can be made open in a topological realization. We now consider the more general problem of which n-ary relations for $n \ge 2$ can be made open. For G-invariant relations, this amounts to topological realization of standard Borel relational G-structures in the sense of first-order logic. For ease of notation, the following discussion will focus on $n = 2$ .

For standard Borel G-spaces $X, Y$ , we have the following analogous characterization. We adopt the following convention: $\alpha _X \times \alpha _Y$ will denote the product action $G^2 \times X \times Y \to X \times Y$ of $G^2$ (i.e., we silently swap the middle two variables of the product map $G \times X \times G \times Y \to X \times Y$ ), while $\alpha _{X \times Y}$ will denote the diagonal action $G \times X \times Y \to X \times Y$ . Note that $\alpha _{X \times Y}$ factors through $\alpha _X \times \alpha _Y$ via the diagonal $G \to G^2$ .

Proof. First, note that in (ii), we indeed have

(3.5.3) $$ \begin{align} (U \times V) * (A \times B) = (U * A) \times (V * B) \end{align} $$

for $U, V \in \mathcal {B}(G)$ , $A \in \mathcal {B}(X)$ , and $B \in \mathcal {B}(Y)$ since

$$ \begin{align*} (U * A) \times (V * B) &= \exists^*_{\alpha_X}(U \times A) \times \exists^*_{\alpha_Y}(V \times B) \\ &= \exists^*_{\alpha_X \times Y}(U \times A \times \exists^*_{\alpha_Y}(V \times B)) &&\text{by Beck--Chevalley (2.3.7)} \\ &= \exists^*_{\alpha_X \times Y}(\exists^*_{G \times X \times \alpha_Y}(U \times V \times A \times B)) &&\text{by Beck--Chevalley (2.3.7)} \\ &= \exists^*_{\alpha_X \times \alpha_Y}(U \times V \times A \times B) &&\text{by Kuratowski--Ulam Theorem}~{2.4.8} \\ &= (U \times V) * (A \times B) \end{align*} $$

(where as indicated above, we silently switch the middle two factors in $G \times X \times G \times Y$ ), whence $\mathcal {O}(G^2) \circledast (\mathcal {S}(X) \otimes \mathcal {S}(Y)) = (\mathcal {O}(G) \circledast \mathcal {S}(X)) \otimes (\mathcal {O}(G) \circledast \mathcal {S}(Y))$ ; as before, this is also equal to $\mathcal {B}(G^2) \circledast (\mathcal {S}(X) \otimes \mathcal {S}(Y))$ by (3.2.9).

In particular, from the assumptions $\mathcal {O}(G) \circledast \mathcal {S}(X) \subseteq \mathcal {S}(X)$ and $\mathcal {O}(G) \circledast \mathcal {S}(Y) \subseteq \mathcal {S}(Y)$ , we get $\mathcal {O}(G^2) \circledast (\mathcal {S}(X) \otimes \mathcal {S}(Y)) \subseteq \mathcal {S}(X) \otimes \mathcal {S}(Y)$ . Now, (i) clearly implies Theorem 3.3.2(i) for $G^2 \curvearrowright X \times Y$ , which by Theorem 3.3.2 is equivalent to each of (ii)–(v). And given countably many R as in (ii), as well as countably many $A \subseteq X$ and $B \subseteq Y$ as in Theorem 3.3.2, by that result, we may find topologies on $X, Y$ making each of these $A, B$ as well as the sets $U_i * A_i$ and $V_i * B_i$ in (ii) open, whence R is open as in (i). This proves the equivalence of (i)–(v).

Since $\alpha _{X \times Y}$ factors through $\alpha _X \times \alpha _Y$ , we have (v) $\implies $ (xi).

Since $\mathcal {O}(G) \otimes \mathcal {S}(X) \subseteq (\mathcal {O}(G) \otimes \mathcal {S}(X))^*_{\pi _2}$ , and similarly for Y, (vii) $\implies $ (vi) and (xi) $\implies $ (viii).

As usual, we have (xi) $\implies $ (vii) and (viii) $\implies $ (vi) because $R = \exists ^*_{\alpha _{X \times Y}}(\alpha _{X \times Y}^{-1}(R))$ .

Finally, we prove (vi) $\implies $ (ii). Let $D \in (\mathcal {O}(G) \otimes \mathcal {S}(X))^*_{\pi _2}$ and $E \in (\mathcal {O}(G) \otimes \mathcal {S}(Y))^*_{\pi _2}$ , say $D =^*_{\pi _2} \bigcup _i (U_i \times A_i) \in \mathcal {O}(G) \otimes \mathcal {S}(X)$ and $E =^*_{\pi _2} \bigcup _j (V_j \times B_j) \in \mathcal {O}(G) \otimes \mathcal {S}(Y)$ . Note that

1. (3.5.4) If $M \subseteq G$ is meager, then $M \times G, G \times M \subseteq G^2$ are orbitwise meager for the diagonal action $G \curvearrowright G^2$ since $\alpha _{G^2}^{-1}(M \times G) = \mu ^{-1}(M) \times G \subseteq G^3$ is $\pi _{23}$ -fiberwise homeomorphic via $(g,h,k) \mapsto (gh,h,k)$ to $M \times G^2$ .

It follows that

$$ \begin{align*} D \times E = (D \times G \times Y) \cap (G \times X \times E) =^*_G \bigcup_{i,j} (U_i \times V_j \times A_i \times B_j) \in \mathcal{O}(G^2) \otimes \mathcal{S}(X) \otimes \mathcal{S}(Y) \end{align*} $$

(again silently swapping the middle two factors). Thus, by Pettis’s theorem (3.2.8) and (3.2.10), for any $\mathcal {B}(G) \ni W =^* W' \in \mathcal {O}(G)$ , we have

$$ \begin{align*} W * (D \times E) &= \bigcup_{i,j} ((W' * (U_i \times V_j)) \times A_i \times B_j) \in \mathcal{O}(G^2) \otimes \mathcal{S}(X) \otimes \mathcal{S}(Y). \end{align*} $$

But now

$$ \begin{align*} W * (\exists^*_{\alpha_X}(D) \times \exists^*_{\alpha_Y}(E)) \\ &= W * \exists^*_{\alpha_X \times \alpha_Y}(D \times E) &&\text{by Kuratowski--Ulam as in ({3.5.3})} \\ &= {\exists^*_{\alpha_{X \times Y}}(W \times \exists^*_{\alpha_X \times \alpha_Y}(D \times E))} \\ &= \exists^*_{\alpha_X \times \alpha_Y}(\exists^*_{\alpha_{G^2 \times X \times Y}}(W \times D \times E)) &&\text{by Kuratowski--Ulam as in ({3.2.14})} \\ &= \exists^*_{\alpha_X \times \alpha_Y}(W * (D \times E)) \\ &\in {\exists^*_{\alpha_X \times \alpha_Y}(\mathcal{O}(G^2) \otimes \mathcal{S}(X) \otimes \mathcal{S}(Y)) = \mathcal{O}(G^2) \circledast (\mathcal{S}(X) \otimes \mathcal{S}(Y))} \end{align*} $$

satisfies (ii), as desired.

For certain $\mathcal {S}$ , however, we can make such a simplification:

As noted before, these results straightforwardly generalize to n-ary relations for all $n \in \mathbb {N}$ . Rather than state the most general result, which would be notationally rather messy, we will only state the generalized form of the main conditions of Corollary 3.5.6: