Notes on quasiPolish spaces
—¿⊳
1 Introduction
A quasiPolish space is, informally, a wellbehaved topological space which can be thought of as a generalization of a Polish space not required to obey any separation axioms beyond . QuasiPolish spaces were introduced by de Brecht [deB], who showed that they satisfy analogs of many of the basic descriptive settheoretic properties of Polish spaces. QuasiPolish spaces also admit some natural constructions with no good analogs for Polish spaces (e.g., the lower powerspace of closed sets; see Section 9); thus, quasiPolish spaces can be useful to consider even when one is initially interested only in the Polish context.
In [deB], quasiPolish spaces are defined as secondcountable, completely quasimetrizable spaces, where a quasimetric is a generalization of a metric that is not required to obey the symmetry axiom . This is a natural generalization of the definition of Polish spaces as secondcountable, completely metrizable spaces. It is then proved that
[[deB, Theorem 24]] QuasiPolish spaces are precisely the homeomorphic copies of subsets of .
Here is the Sierpinski space, with open but not closed, and can be thought of as the topological space with a “generic” open set (namely ). Similarly, the product can be thought of as the space with countably many “generic” open sets (the subbasic ones). In nonmetrizable spaces such as , sets are not so wellbehaved since they may not include all closed sets; thus it is convenient to alter the classical definition of to mean all sets
for countably many open sets . Note that the above set can be read as “the set of all where the implications hold”. Thus, Section 1 can be read as
“space with countably many generic open sets, and  
The purpose of these notes is to give a concise, selfcontained account of the basic theory of quasiPolish spaces from this point of view. That is, we take Section 1 as a definition; in fact, we will not mention quasimetrics at all. Whenever we show that a space is quasiPolish, we will give an explicit definition of it as a subspace of a known quasiPolish space (such as ). Our exposition also makes no reference to domain theory or various other classes of spaces inspired by computability theory (see e.g., [deB, §9]). It is hoped that such an approach will be easily accessible to descriptive set theorists and others familiar with the classical theory of Polish spaces.
We would like to stress that these notes contain essentially no new results. Most of the results that follow are from the papers [deB] and [dBK], or are easy generalizations of classical results for Polish spaces. Whenever possible, we give a reference to the same (or equivalent) result in one of these papers. However, the proofs we give are usually quite different from those referenced, reflecting our differing point of view.
As our main goal is to give a concise exposition of the basic results about quasiPolish spaces, we have neglected to treat many other relevant topics, e.g., local compactness [deB, §8], the Hausdorff–Kuratowski theorem and difference hierarchy [deB, §13], Hurewicz’s theorem for nonquasiPolish sets [dB2], and upper powerspaces [dBK], among others. For the same reason, we do not include a comprehensive bibliography, for which we refer the reader to the aforementioned papers.
Finally, we remark that our approach is heavily inspired by the correspondence between quasiPolish spaces and countably (co)presented locales [Hec]. A locale is, informally, a topological space without an underlying set, consisting only of an abstract lattice of “open sets”. The definition of quasiPolish spaces in terms of countably many “generators and relations” for their open set lattices leads naturally to the idea of forgetting about the points altogether and regarding the open sets as an abstract lattice, i.e., replacing spaces with locales. In what follows, we will not refer explicitly to the localic viewpoint; however, the reader who is familiar with locale theory will no doubt recognize its influence in several places (most notably Section 8).
Acknowledgments. We would like to thank Alexander Kechris and Matthew de Brecht for providing some comments on earlier revisions of these notes.
2 Basic definitions
Recall that on an arbitrary topological space , the specialization preorder is given by
The specialization preorder is a partial order iff is , and is discrete iff is . Open sets are upwardclosed; closed sets are downwardclosed. The principal ideal
generated by a point coincides with its closure .
The Sierpinski space has open but not closed; the specialization order is thus given by .
We will be concerned with product spaces and their subspaces, especially for countable. Whenever convenient, we identify with , the powerset of ; note that the specialization order on corresponds to inclusion of subsets. A basis of open sets in consists of the sets
Given an arbitrary topological space , not necessarily metrizable, we define the Borel hierarchy on as follows; this definition is due to Selivanov [Sel]. The sets are the open sets. For an ordinal , the sets are those of the form
(we write for the set of sets in ). It is easy to see by induction that for , we may take above, as in the usual definition of the Borel hierarchy (in the metrizable case). The sets are the complements of the sets, and the sets are those which are both and ; these are denoted respectively. A set is Borel if it is for some . We have the usual picture of the Borel hierarchy:
Of particular note are the sets
for open; they are the result of “imposing countably many relations between open sets”. The following are immediate:
[[deB, Proposition 8]] Points in a firstcountable space are . ∎
[[deB, Proposition 9]] The specialization preorder on a secondcountable space is . Hence, the equality relation on a secondcountable space is . ∎
A quasiPolish space is a homeomorphic copy of a subspace of for some countable , equivalently of [deB, Theorem 24]. In other words, it is the result of imposing countably many relations between countably many “generic” open sets (the subbasic open sets ). This is made more explicit by the following definitions.
For a topological space and a collection of open sets in , define
is continuous, and is an embedding if is and is a subbasis, in which case we call the canonical embedding (with respect to ). A countable copresentation of a space consists of a countable subbasis for together with a definition of . Thus, is quasiPolish iff it is countably copresented (has a countable copresentation).
Many properties of quasiPolish spaces can also be established with no extra effort for the more general class of countably correlated spaces, which are homeomorphic copies of subspaces of for arbitrary index sets .
Recall that a Polish space is a separable, completely metrizable topological space, while a standard Borel space is a set equipped with the Borel algebra of some Polish topology. See [Kec] for basic descriptive set theory on Polish spaces. We will show below (Footnote 1) that quasiPolish spaces are a generalization of Polish spaces; hence, most of the results that follow are generalizations of their classical analogs for Polish spaces.
3 Basic properties
{proposition}QuasiPolish spaces are standard Borel, and can be made Polish by adjoining countably many closed sets to the topology.
Proof.
If is , then is , and is the result of adjoining the complements of the (sub)basic open sets in (whence the Borel algebras agree). ∎
[[deB, Theorem 22]] A subspace of a quasiPolish space is quasiPolish. Similarly for countably correlated spaces.
Proof.
Obvious. ∎
[[deB, Corollary 43]] A countable product of quasiPolish spaces is quasiPolish. Similarly for countably correlated spaces.
Proof.
If are , then so is where is the th projection. ∎
For any topological space , let
where the open sets are those in together with . (Thus, is a newly adjoined least element in the specialization preorder, often thought of as “undefined”.)
If is quasiPolish, then so is . Similarly for countably correlated spaces.
Proof.
Suppose is . Then
[[deB, Corollary 43]] A countable disjoint union of quasiPolish spaces is quasiPolish. Similarly for countably correlated spaces.
Proof.
Let be quasiPolish (or countably correlated). Then
A topological space is locally quasiPolish if it has a countable cover by open quasiPolish subspaces.
Locally quasiPolish spaces are quasiPolish.
Proof.
Let be as above. Then
4 Subspaces
Recall [Kec, 3.11] that a subspace of a Polish space is Polish iff it is . An analogous fact holds for quasiPolish spaces.
[[deB, Theorem 21]] Let be a secondcountable space and be a countably correlated subspace. Then is .
The proof we give consists essentially of applying the following simple fact in universal algebra to the lattice of open sets of . Given any countably presented algebraic structure (e.g., group, ring, …) and countably many generators , there is a countable presentation of using only those generators. To see this: let be any countable presentation; write each as some word in the , and substitute into to get a set of relations in the ; write each as some word in the , and substitute into to get a word in the (which evaluates to in ); then .
Proof.
Let be an embedding with image, say where are open. Thus each is a union of basic open sets:
where runs over finite subsets of . For each , let be open such that
For each finite , put , so that
Let be a countable subbasis of open sets in . We claim that
is straightforward. To prove , let belong to the righthand side. Put
Using , we easily have . Let . For each , we have
Thus since is , . ∎
[[deB, Theorem 23]] Let be a quasiPolish space. A subspace is quasiPolish iff it is . ∎
A space is quasiPolish iff it is secondcountable and countably correlated.
5 Polish spaces
{theorem}[[deB]^{1}^{1}1Since [deB] defines quasiPolish spaces in terms of complete quasimetrics, which generalize complete metrics, Footnote 1 is trivial according to the definitions in [deB]. In fact, the content of Footnote 1 is contained in the proofs of [deB, Theorems 19–21] (which establish that their definition of quasiPolish space implies ours).] Polish spaces are quasiPolish.
Proof 1.
First, we note
is quasiPolish.
Proof.
We have
Now let be a Polish space with compatible complete metric and be a countable dense subset. Then using a standard construction of the completion of ,
Sometimes it is useful to have a countable copresentation of a Polish space derived from a countable basis instead of a countable dense subset (as in the above proof). This is provided by the following alternative proof, which is also more direct in that it avoids first showing that is quasiPolish.
Proof 2.
Let be a Polish space with compatible complete metric . Let be a countable basis of open sets in , closed under binary intersections (so containing ). For and , put
the neighborhood of . We claim that the canonical embedding has image
() 
is straightforward. To prove , let belong to the righthand side; we must find such that for all . By the first three conditions on the righthand side (), is a Cauchy filter base. Let be its limit, i.e.,
For such that , since is Cauchy, there is some such that , whence by the second condition on the righthand side (). Conversely, for , by the fourth condition on the righthand side () there is some and with , whence . ∎
A topological space is Polish iff it is quasiPolish and regular.
Proof.
If is quasiPolish and regular, then is secondcountable and , whence by the Urysohn metrization theorem, is metrizable; letting be a completion of with respect to a compatible metric, is Polish, and is by Section 4, hence Polish. ∎
6 Change of topology
{theorem}[[deB, Theorem 73]] Let be a quasiPolish space and be countably many sets. Then the space given by with adjoined to its topology is quasiPolish. Similarly for countably correlated spaces.
Proof.
We have
As noted in [deB, paragraph before Lemma 72], given a Polish space, adjoining sets which are not closed might result in a nonmetrizable space.
We also have a converse to Section 6 in the case of a single set:
Let be a quasiPolish space and be such that the space given by with adjoined to its topology is quasiPolish. Then is .
Proof.
[[deB, Lemma 72]] Let be a quasiPolish space and be finer quasiPolish topologies on . Then the topology generated by is quasiPolish.
Proof.
We have
[[deB, Theorem 74]] Let be a quasiPolish space and . Then there is a finer quasiPolish topology on containing each and contained in .
Proof.
By Section 6 it suffices to consider the case of a single . We induct on . The case is trivial, so assume . Write where for . By the induction hypothesis, there are finer quasiPolish topologies such that . Then each , so by Section 6, the topology generated by and is quasiPolish. Now by Section 6, the topology generated by the is quasiPolish. Clearly , whence ; and . ∎
7 Baire category
Recall [Kec, §8] that a topological space is Baire if the intersection of countably many dense open sets in is dense; and that a subset is comeager if it contains a countable intersection of dense open sets, meager if its complement is comeager, and Bairemeasurable (or has the Baire property) if it differs from an open set by a meager set.
In the nonmetrizable setting, it is useful to note the following:
Let be a topological space, be a dense subset. Then is comeager. Thus, is comeager iff it contains a countable intersection of dense sets.
Proof.
Let where are open. Since is dense, so is each , i.e., ; since is closed, this implies . So the are dense open sets whose intersection is contained in . ∎
A space is completely Baire if every closed subspace is Baire.
Let be a topological space. The following are equivalent:

Every subspace is Baire.

is completely Baire.

Every nonempty closed is nonmeager in .
Proof.
Clearly (i)(ii)(iii). Assume (iii), and let be ; we show that is Baire. Let be open sets dense in ; we must show that is dense in . Let be open with ; we must show that . Put ; clearly . Since is and dense in , by Section 7 there are dense open in with . Each is dense in , hence also in , so by (iii), , as desired. ∎
[Baire category theorem [deB, Corollary 52]] Countably correlated spaces are (completely) Baire.
Proof.
By Section 7, it is enough to show that every nonempty closed is nonmeager in . Let be open and dense in ; we must show that . We will find finite and . Let ; then , so there is some . Given such that , since is dense in , we have , so there is some , whence there is some basic open such that , whence and . Put . Then for each , and , whence , as desired. ∎
As for Polish spaces [Kec, §8.J], we also have a wellbehaved theory of “fiberwise” Baire category, i.e., category quantifiers, for quasiPolish spaces. We will state this in a more general context.
Let be a function between sets , such that for each , the fiber is equipped with a topology. For a subset , put
A subset is fiberwise open if is open in for each ; notions such as fiberwise Baire, fiberwise Bairemeasurable are defined similarly. A family of fiberwise open subsets of is a fiberwise weak basis for a fiberwise open if for every and nonempty open , there is some with .
[see [Kec, 8.27]] Let be as above.

If is fiberwise Baire, then for fiberwise open ,

For countably many ,

If is fiberwise Baire, then for fiberwise open , fiberwise Bairemeasurable , and a fiberwise weak basis for ,
Proof.
(i) and (ii) are straightforward. For (iii), if , i.e., is nonmeager in , then letting (by the Baire property) where is open and is meager, we have some with , whence , and , whence . Conversely, if with , i.e., but is meager in , then is nonmeager (since is Baire), i.e., . ∎
The following result generalizes the wellknown fact [Kec, 22.22] that category quantifiers applied to Borel sets in products of Polish spaces preserve Borel complexity.
Let be a continuous open map, where is a secondcountable completely Baire space. Then is fiberwise Baire, and for every , we have .
Proof.
Since is secondcountable, so is , whence points are , whence fibers for are , hence Baire. Let be a countable basis of open sets in ; then is a fiberwise weak basis for any open . So the hypotheses of Section 7 are satisfied. Now induct on , using Section 7 and the fact that for , consists precisely of sets of the form with open and , . ∎
We also have the following generalization of the classical Kuratowski–Ulam theorem [Kec, 8.41]; the proof is essentially from [MT, A.1]. Recall that a continuous map is categorypreserving if the preimage of every meager set is meager; this includes all open maps.
[Kuratowski–Ulam theorem] Let be a continuous open map, where is a secondcountable completely Baire space. Then for every Bairemeasurable ,

is Bairemeasurable in for comeagerly many ;

are Bairemeasurable;

(respectively ) is (co)meager iff is.
Proof.
First, we show in (iii). Let be comeager. By Section 7(ii), we may assume is dense open. Let be a countable basis of open sets in . Then for each , is dense open in , since if is open then whence (since is dense) whence . It follows that
is (a countable intersection of dense sets, hence) comeager. We have iff for every with we have , i.e., iff is dense in . Thus , and so is comeager, as desired.
Now let be Bairemeasurable, say where is open and is meager. Then for all of the comeagerly many (by in (iii)), we have that is meager in , whence is Bairemeasurable, proving (i), and is comeager (or meager) in iff is. The latter implies that is meager; by Section 7, is and so Bairemeasurable, whence is Bairemeasurable, proving (ii). Similarly, is meager. Now to prove in (iii): if is meager, then so is (by Section 7(i)), whence so is since is categorypreserving, whence so is . ∎
We close this section with some simple applications of Baire category.
Let be a continuous open map between quasiPolish spaces. Then for any and fiberwise open , is .
Let be a continuous open surjection between quasiPolish spaces. Then is iff is.
Proof.
Since is surjective, , which is by Section 7 if is. ∎
Let be a continuous open map between quasiPolish spaces. Then admits a Borel section , i.e., a Borel map such that .
Proof.
Apply the large section uniformization theorem [Kec, 18.6] to the inverse graph relation of , , using the ideals
for each . Clearly each fiber is or ; and is BorelonBorel (see [Kec, 18.5]), since for every (quasi)Polish space and Borel set , we have
(where takes to ), which is Borel in by Section 7. It follows that has a Borel uniformizing function , which is the desired section. ∎
A topological space is irreducible if , and whenever with closed, then either or . A topological space is sober if is , and for every irreducible closed , there is a (unique, by ) such that .
[[deB, Corollary 39]] QuasiPolish spaces are sober.
Proof.
Let be quasiPolish and be irreducible closed. Let be a countable subbasis of open sets in . For every open which both intersect , by irreducibility, also . Thus for every such that , is dense. So by Baire category, there is some , which is easily seen to satisfy . ∎
8 Posites
In this section, we study a special kind of copresentation, one where all of the relations between open sets are of the form “open sets cover ”.
A posite^{2}^{2}2This notion comes from locale theory; see [Joh, II 2.11]. consists of a poset and a binary relation between subsets of and elements of . We think of elements as names for basic open sets, and of the relation for as meaning “ cover ”. The relation is required to satisfy:
(8.1)  
(8.2) 
(the second condition says “every open cover of refines to an open cover of ”).
Every posite determines a topological space, as follows. For a poset , let
denote the space of upwardclosed subsets of , and let
denote the space of filters in . Now for a posite , let
denote the space of coideals in , i.e., the complements of ideals , which are downwardclosed subsets such that . Finally, let
denote the space of prime filters in ; we call the space copresented by . We think of as a “point”, where are the “basic neighborhoods” of .
A posite is countable if both and