Baire Spaces and the Baire Spaces and the Wijsman TopologyWijsman Topology

Jiling CaoJiling Cao

Auckland University of Technology Auckland University of Technology

New ZealandNew Zealand

This is joint work with This is joint work with

Artur H. TomitaArtur H. Tomita

SSãão Paulo University, Brazilo Paulo University, Brazil

International Conference on Topology International Conference on Topology

and its Applications 2007 and its Applications 2007

December 3-7, Kyoto, JapanDecember 3-7, Kyoto, Japan

What is the Wijsman topology?What is the Wijsman topology?

It originates in the study of convergence of convex sets in It originates in the study of convergence of convex sets in Euclidean space by Euclidean space by R. WijsmanR. Wijsman in 1960s. The general in 1960s. The general setting was developed in 1980s.setting was developed in 1980s.

Let (Let (XX,,dd)) be a metric space. If be a metric space. If xx is a point in is a point in XX,, andand A A is a is a nonempty closed subset of nonempty closed subset of XX::

xx

AA

The The gapgap between between AA and and xx is: is: d d((AA,,xx) = inf ) = inf {{dd((aa,,xx): ): aa in in AA}.}.

2X

DefineDefine

FF((XX) = {all nonempty closed subset of ) = {all nonempty closed subset of X X }. }.

If If AA runs through runs through FF((XX) and ) and xx runs through runs through XX, we have a , we have a two-variable mapping:two-variable mapping: dd((··,,··):): F F((XX) ) xx XX → → R.R.

For any fixed For any fixed AA in in FF((XX)),, it is well-known (and simple) that it is well-known (and simple) that

dd((AA,,··):): ((XX,,dd) → () → (RR,,TTuu) is a continuous mapping. ) is a continuous mapping.

What will happen if we consider What will happen if we consider dd((··,, x x):): F F((XX) → () → (RR,,TTuu) for ) for allall fixed fixed x x inin X X ? ?

The The Wijsman topology Wijsman topology TTwdwd on on FF((XX), is the weakest topology ), is the weakest topology such that for each such that for each xx in in XX, the mapping , the mapping dd((··,,xx)) isis continuous. continuous.

By definition, a subbase for the Wijsman topology is By definition, a subbase for the Wijsman topology is

In the past 40 years, this topology has been extensively In the past 40 years, this topology has been extensively investigated. For more details, refer to the book, “investigated. For more details, refer to the book, “Topologies Topologies on closed and closed convex setson closed and closed convex sets”, by G. Beer in 1993.”, by G. Beer in 1993.

Some basic propertiesSome basic properties

The Wijsman topology is The Wijsman topology is notnot a topological invariant. a topological invariant.

For each For each AA in in FF((XX)) , A, A ↔↔ dd((AA,,··):): ((XX,,dd) → () → (RR,,TTuu). Then, ). Then, ((FF((XX),),TTwdwd) embeds in ) embeds in CCpp((XX) ) as a subspace.as a subspace. Thus, Thus, ((FF((XX),),TTwdwd) is a ) is a Tychonoff spaceTychonoff space. .

Theorem Theorem For a metric space (For a metric space (XX,,dd), the following are ), the following are equivalent:equivalent:(1) ((1) (FF((XX),),TTwdwd) is metrizable;) is metrizable;

(2) (2) ((FF((XX),),TTwdwd) is second countable;) is second countable;

(3) ((3) (FF((XX),),TTwdwd) is first countable;) is first countable;(4) (4) XX is separable. is separable.(Cornet, Francaviglia, Lechicki, Levi)(Cornet, Francaviglia, Lechicki, Levi)

..

Completeness properties Completeness properties

TheoremTheorem (Effros, 1965): If (Effros, 1965): If XX is a is a Polish Polish space, then there space, then there is a compatible metric is a compatible metric dd on on XX such that ( such that (FF((XX),),TTwdwd) is ) is Polish.Polish.

TheoremTheorem (Beer, 1991): If ( (Beer, 1991): If (X,dX,d) is a complete and ) is a complete and separable metric space, then (separable metric space, then (FF((XX),),TTwdwd) is Polish.) is Polish.

TheoremTheorem (Constantini, 1995): If (Constantini, 1995): If XX is a Polish space, then is a Polish space, then ((FF((XX),),TTwdwd) is Polish for every compatible metric ) is Polish for every compatible metric dd on on X.X.

ExampleExample (Constantini, 1998): There is a complete metric (Constantini, 1998): There is a complete metric space (space (XX,,dd) such that () such that (FF((XX),),TTwdwd) is not ) is not ČČech-completeech-complete..

Baire spacesBaire spaces

A space A space XX is is BaireBaire if the intersection of each sequence of if the intersection of each sequence of dense open subsets of dense open subsets of XX is dense in is dense in XX; and a Baire ; and a Baire space space XX is called is called hereditarily Bairehereditarily Baire if each nonempty if each nonempty closed subspace of closed subspace of XX is Baire. is Baire.

Baire Category TheoremBaire Category Theorem: Every : Every ČČech-complete space is ech-complete space is hereditarily Baire.hereditarily Baire.

Concerning Baireness of the Wijsman topology, we have Concerning Baireness of the Wijsman topology, we have the following known result:the following known result:

TheoremTheorem (Zsilinszky, 1998): If ( (Zsilinszky, 1998): If (XX,,dd) is a complete metric ) is a complete metric

space, then (space, then (FF((XX),),TTwdwd) is Baire.) is Baire.

TheoremTheorem (Zsilinszky, 20??): Let (Zsilinszky, 20??): Let XX be an almost locally be an almost locally separable metrizable space. Then separable metrizable space. Then XX is Baire if and only if is Baire if and only if

((FF((XX),),TTwdwd) is Baire for each compatible metric ) is Baire for each compatible metric dd on on X.X.

A space is A space is almost locally separablealmost locally separable, provided that the set of , provided that the set of points of local separability is dense. points of local separability is dense.

In a metrizable space, this is equivalent to having a In a metrizable space, this is equivalent to having a countable-countable-in-itselfin-itself ππ-base. -base.

CorollaryCorollary (Zsilinszky, 20??): A space (Zsilinszky, 20??): A space X X is separable, metrizable is separable, metrizable

and Baire if and only if (and Baire if and only if (FF((XX),),TTwdwd) is a metrizable and Baire ) is a metrizable and Baire space for each compatible metric space for each compatible metric dd on on X.X.

Motivated by the previous results, Zsilinszky posed two open Motivated by the previous results, Zsilinszky posed two open questions. The first one is:questions. The first one is:

Question 1Question 1 (Zsilinszky, 1998): Suppose that ( (Zsilinszky, 1998): Suppose that (X,dX,d) is a ) is a

complete metric space. Must (complete metric space. Must (FF((XX),),TTwdwd) be hereditarily ) be hereditarily Baire? Baire?

Therefore, the answer to Question 1 is Therefore, the answer to Question 1 is negativenegative..

TheoremTheorem (Chaber and Pol, 2002): Let (Chaber and Pol, 2002): Let XX be a metrizable be a metrizable space such that the set of points in space such that the set of points in XX without any compact without any compact neighbourhood has weight . Then for any compatible neighbourhood has weight . Then for any compatible

metric metric dd on X, embeds in ( on X, embeds in (FF((XX),),TTwdwd) as closed ) as closed subspace. In particular, the Wijsman hyperspace contains a subspace. In particular, the Wijsman hyperspace contains a closed copy of rationals.closed copy of rationals.

The second question was posed by Zsilinszky at the 10The second question was posed by Zsilinszky at the 10thth Prague Toposym in 2006. Prague Toposym in 2006.

Question 2Question 2 (Zsilinszky, 2006): Suppose that ( (Zsilinszky, 2006): Suppose that (X,dX,d) is a metric ) is a metric

hereditarily Baire space. Must (hereditarily Baire space. Must (FF((XX),),TTwdwd) be Baire?) be Baire?

The main purpose of this talk is to present an The main purpose of this talk is to present an affirmativeaffirmative answer answer to Question 2.to Question 2.

To achieve our goal, we shall need three tools:To achieve our goal, we shall need three tools:

The ball proximal topology on The ball proximal topology on FF((XX).). The pinched-cube topologyThe pinched-cube topology on the countable power of on the countable power of XX..

A game-theoretic characterization of Baireness.A game-theoretic characterization of Baireness.

The ball proximal topology The ball proximal topology

For each open subset For each open subset EE of of XX, we define, we define

The The ball proximal topologyball proximal topology TTbpdbpd on on FF((XX)) has a subbase has a subbase

where where

In general,In general, TTwdwd is coarser than but is coarser than but different from different from TTbpd bpd on on

FF(X)(X). . Thus,Thus, the identity mapping the identity mapping ii: (: (FF((XX),),TTbpdbpd) → () → (FF((XX),),TTwdwd) ) is is continuouscontinuous, but is , but is not opennot open in general. in general.

For each nonempty open subset For each nonempty open subset UU in ( in (FF((XX),),TTbpdbpd), ), ii(U) has (U) has nonempty interior in (nonempty interior in (FF((XX),),TTwdwd). ).

A mapping A mapping ff:: X → Y X → Y is called is called feebly open feebly open if if ff((UU) has nonempty ) has nonempty interior for every nonempty open set interior for every nonempty open set UU of of XX. .

Similarly, a mapping Similarly, a mapping ff:: X → Y X → Y is called is called feebly continuous feebly continuous whenever the pre-image of an open set is nonempty, the pre-whenever the pre-image of an open set is nonempty, the pre-image has a nonempty interior. image has a nonempty interior.

TheoremTheorem (Frolik, 1965; Neubrunn, 1977): Let (Frolik, 1965; Neubrunn, 1977): Let ff:: X → Y X → Y be a be a feebly continuous and feebly open bijection. Then feebly continuous and feebly open bijection. Then XX is Baire if is Baire if and only if and only if YY is Baire. is Baire.

TheoremTheorem (Zsilinszky, 20??): Let ( (Zsilinszky, 20??): Let (X,dX,d) be a metric space. ) be a metric space. ThenThen ((FF((XX),),TTbpdbpd) ) is Baire if and only ifis Baire if and only if ( (FF((XX),),TTwdwd)) is Baire.is Baire.

The pinched-cube topologyThe pinched-cube topology

The idea of pinched-cube topologies originates in McCoy’s The idea of pinched-cube topologies originates in McCoy’s work in 1975.work in 1975.

The term was first used by Piotrowski, RosThe term was first used by Piotrowski, Rosłanowski and łanowski and ScottScott in 1983. in 1983.

Let (Let (X,dX,d) be a metric space. Define) be a metric space. Define

∆∆dd = = {{finite unions of proper closed balls in finite unions of proper closed balls in XX}}..

For For BB in in ∆∆dd,, and and UUii (i < n) (i < n) disjoint from disjoint from BB, we define, we define

The collection of sets of this form is a base for a topology The collection of sets of this form is a base for a topology TTpdpd on , called the on , called the pinched-cube topologypinched-cube topology..

Theorem Theorem (Cao and Tomita):(Cao and Tomita): Let (Let (X,dX,d) be a metric space. If ) be a metric space. If

with with TTpdpd is Baire, then ( is Baire, then (FF((XX),),TTbpdbpd) is Baire.) is Baire.

Let be the subspace of (Let be the subspace of (FF((XX),),TTbpdbpd) consisting of all ) consisting of all separable subsets of separable subsets of XX. .

Define by lettingDefine by letting

It can be checked that It can be checked that ff is continuous, and feebly open. is continuous, and feebly open.

Then, it follows that (Then, it follows that (FF((XX),),TTbpdbpd) is Baire.) is Baire.

Now, we consider the Choquet game played in Now, we consider the Choquet game played in XX. .

There are two players, There are two players, ββ and and αα, who select nonempty , who select nonempty open sets in open sets in XX alternatively with alternatively with ββ starting first: starting first:

We claim that We claim that αα wins wins a play if a play if

Otherwise, we say that Otherwise, we say that ββ has wonhas won the play already. the play already.

Without loss of generality, we can always restrict moves Without loss of generality, we can always restrict moves of of ββ andand αα in a base or in a base or ππ-base of -base of XX..

The Choquet gameThe Choquet game

Baireness of a space can be characterized by applying the Baireness of a space can be characterized by applying the Choquet game.Choquet game.

Theorem Theorem (Oxtoby, Krom, Saint-Raymond): A space (Oxtoby, Krom, Saint-Raymond): A space XX is not is not a Baire space if and only if a Baire space if and only if ββ has a winning strategy.has a winning strategy.

This theorem was first given by J. Oxtoby in 1957. Then, it This theorem was first given by J. Oxtoby in 1957. Then, it was re-discovered by M. R. Krom in 1974, and later on by was re-discovered by M. R. Krom in 1974, and later on by J. Saint-Raymond in 1983.J. Saint-Raymond in 1983.

By a By a strategystrategy for the player for the player ββ, we mean a mapping defined , we mean a mapping defined for all finite sequences of moves of for all finite sequences of moves of αα. A . A winning strategywinning strategy for for ββ is a strategy which can be used by is a strategy which can be used by ββ to win each play no to win each play no matter how matter how αα moves in the game. moves in the game.

TheoremTheorem (Cao and Tomita): Let ( (Cao and Tomita): Let (XX,,dd) be a metric space. If ) be a metric space. If with the Tychonoff topology is Baire, then with with the Tychonoff topology is Baire, then with TTpdpd is is also Baire.also Baire.

The basic idea is to apply the game characterization of The basic idea is to apply the game characterization of Baireness.Baireness.

Suppose that with the Tychonoff topology is Baire. Pick up Suppose that with the Tychonoff topology is Baire. Pick up any strategy any strategy σσ for for ββ in in TTpd pd ..

We apply We apply σσ to construct inductively a strategy to construct inductively a strategy ΘΘ for for ββ in in the Tychonoff topology. Since the Tychonoff topology. Since ΘΘ is not a wining strategy for is not a wining strategy for ββ in the Tychonoff topology, it can be show that in the Tychonoff topology, it can be show that σσ is not a is not a winningwinning for for ββ inin TTpd pd either.either.

Main resultsMain results

TheoremTheorem (Cao and Tomita): Let ( (Cao and Tomita): Let (XX,,dd) be a metric and ) be a metric and hereditarily Baire space. Then (hereditarily Baire space. Then (FF((XX),),TTwdwd) is Baire. ) is Baire.

Note that the Tychonoff product of any family of Note that the Tychonoff product of any family of metric hereditarily Baire spaces is Baire (Chaber and Pol, metric hereditarily Baire spaces is Baire (Chaber and Pol, 2005). 2005).

It follows that with the Tychonoff topology is Baire, andIt follows that with the Tychonoff topology is Baire, and

thus with thus with TTpdpd is Baire.is Baire.

By one of the previous theorems, (By one of the previous theorems, (FF((XX),),TTpdpd) is a Baire ) is a Baire space. space.

Finally, by another previous theorem, (Finally, by another previous theorem, (FF((XX),),TTwdwd) is Baire. ) is Baire.

Comparison and examplesComparison and examples

For a space For a space XX, recall that the, recall that the Vietoris topology Vietoris topology TTvv on on FF((XX) has ) has

as a base, whereas a base, where

The relationship between the Vietoris topology and the The relationship between the Vietoris topology and the Wijsman topologies is given in the next result.Wijsman topologies is given in the next result.

TheoremTheorem (Beer, Lechicki, Levi, Naimpally, 1992): Let (Beer, Lechicki, Levi, Naimpally, 1992): Let XX be a metrizable space. Then, on be a metrizable space. Then, on FF((XX), ),

TTvv = sup = sup{{ TTwdwd: : dd is a compatibe metric on is a compatibe metric on XX}}..

For a metric space (For a metric space (X,dX,d), we have the following diagram:), we have the following diagram:

BaireBaire

((FF((XX),),TTVV) Baire) Baire ( (FF((XX),),TTwdwd) Baire) Baire

BaireBaire

ExampleExample (Pol, 20??): There is a separable metric ( (Pol, 20??): There is a separable metric (XX,,dd) ) which is of the first Baire category such that (which is of the first Baire category such that (FF((XX),),TTwdwd) is ) is Baire.Baire.

Consider equipped with the metric defined byConsider equipped with the metric defined by

Let be the subspace consisting of sequences which Let be the subspace consisting of sequences which are eventually equal to zero. This subspace is of the first are eventually equal to zero. This subspace is of the first Baire category. Baire category.

The product space is a separable metric The product space is a separable metric space which is of the first Baire category.space which is of the first Baire category.

It can be shown that (It can be shown that (FF((XX),),TTwdwd) is Baire.) is Baire.

ExampleExample (Cao and Tomita): There is a metric Baire space (Cao and Tomita): There is a metric Baire space

((X,dX,d) such that () such that (FF((XX),),TTwdwd) is Baire, but with the ) is Baire, but with the Tychonoff topology is not Baire.Tychonoff topology is not Baire.

Let be a family of disjoint Let be a family of disjoint stationarystationary subsets of subsets of . For each , we put . For each , we put . Let be the metric space defined before. . Let be the metric space defined before. On , we consider the metric defined byOn , we consider the metric defined by

Then, the required space will beThen, the required space will be

equipped with the product metric from .equipped with the product metric from .

Thank YouThank You