A topological space is said to be a Baire space if for every countable family
of open and dense subsets of
, the intersection
is dense in
(equivalently if every nonempty open subset of
is of second category in
). By the Baire category theorem, every complete metric space is a Baire space. The Baire property (i.e. being a Baire space) can be characterized using the Banach-Mazur game, which is the focus of this post.
Baire category theorem and Baire spaces are discussed in this previous post. We define the Banach-Mazur game and show how this game is related to the Baire property. We also define some completeness properties stronger than the Baire property using this game. For a survey on Baire spaces, see [4]. For more information about the Banach-Mazur game, see [1]. Good references for basic topological terms are [3] and [5]. All topological spaces are assumed to be at least Hausdorff.
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The Banach-Mazur Game
The Banach-Mazur game is a two-person game played on a topological space. Let be a space. There are two players,
and
. They take turn choosing nested decreasing nonempty open subsets of
as follows. The player
goes first by choosing a nonempty open subset
of
. The player
then chooses a nonempty open subset
. At the nth play where
,
chooses an open set
and
chooses an open set
. The player
wins if
. Otherwise the player
wins.
If the players in the game described above make the moves , then this sequence of open sets is said to be a play of the game.
The Banach-Mazur game, as described above, is denoted by . In this game, the player
makes the first move. If we modify the game by letting
making the first move, we denote this new game by
. In either version, the goal of player
is to reach an empty intersection of the chosen open sets while player
wants the chosen open sets to have nonempty intersection.
A Remark About Topological Games
Before relating the Banach-Mazur game to Baire spaces, we give a remark about topological games. For any two-person game played on a topological space, we are interested in the following question.
- Can a player, by making his/her moves judiciously, insure that he/she will always win no matter what moves the other player makes?
If the answer to this question is yes, then the player in question is said to have a winning strategy. For an illustration, consider a space that is of first category in itself, so that
where each
is nowhere dense in
. Then player
has a winning strategy in the Banach-Mazur game
. The player
always wins the game by making his/her nth play
.
In general, a strategy for a player in a game is a rule that specifies what moves he/she will make in every possible situation. In other words, a strategy for a player is a function whose domain is the set of all partial plays of the game, and this function tells the player what the next move should be. A winning strategy for a player is a strategy such that this player always wins if that player makes his/her moves using this strategy. A strategy for a player in a game is not a winning strategy if of all the plays of the game resulting from using this strategy, there is at least one specific play of the game resulting in a win for the other player.
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Strategies in the Banach-Mazur Game
With the above discussion in mind, let us discuss the strategies in the Banach-Mazur game. We show that the strategies in this game code a great amount of information about the topological space in which the game is played.
First we discuss strategies for player in the game
. A strategy for player
is a function
such that
(the first move) and for each partial play of the game (
)
,
is a nonempty open set such that
. If player
makes all his/her moves using the strategy
, then the strategy
for player
contains information on all moves of
. We adopt the convention that a strategy for a player in a game depends only on the moves of the other player. Thus for the partial play of the Banach-Mazur game denoted by
above,
.
If is a winning strategy for player
in the game
, then using this strategy will always result in a win for
. On the other hand, if
is a not a winning strategy for player
in the game
, then there exists a specific play of the Banach-Mazur game
such that , and for each
,
and player
wins in this play of the game, that is,
.
In the game (player
making the first move), a strategy for player
is a function
such that for each partial play of the game
,
is a nonempty open set such that
.
We now present a lemma that helps translate game information into topological information.
Lemma 1
Let be a space. Let
be a nonempty open set. Let
be the set of all nonempty open subsets of
. Let
be a function such that for each
,
. Then there exists a disjoint collection
consisting of elements of
such that
is dense in
.
Proof
This is an argument using Zorn’s lemma. If the open set in the hypothesis has only one point, then the conclusion of the lemma holds. So assume that
has at least two points.
Let be the set consisting of all collections
such that each
is a disjoint collection consisting of elements of
. First
. To see this, let
and
be two disjoint open sets such that
and
. This is possible since
has at least two points. Let
. Then we have
. Order
by set inclusion. It is straightforward to show that
is a partially ordered set.
Let be a chain (a totally ordered set). We wish to show that
has an upper bound in
. The candidate for an upper bound is
since it is clear that for each
,
. We only need to show
. To this end, we need to show that any two elements of
are disjoint open sets.
Note that elements of are elements of
. Let
. Then
and
for some
and
. Since
is a chain, either
or
. This means that
and
belong to the same disjoing collection in
. So they are disjoint open sets that are members of
.
By Zorn’s lemma, has a maximal element
, which is a desired disjoint collection of sets in
. Since
is maximal with respect to
,
is dense in
.
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Characterizing Baire Spaces using the Banach-Mazur Game
Lemma 1 is the linkage between the Baire property and the strategies in the Banach-Mazur game. The thickness in Baire spaces and spaces of second category allow us to extract a losing play in any strategy for player . The proofs for both Theorem 1 and Theorem 2 are very similar (after adjusting for differences in who makes the first move). Thus we only present the proof for Theorem 1.
Theorem 1
The space is a Baire space if and only if player
has no winning strategy in the game
.
Proof
Suppose that
is not a Baire space. We define a winning strategy in the game
for player
. The space
not being a Baire space implies that there is some nonempty open set
such that
is of first category in
. Thus
where each
is nowhere dense in
.
We now define a winning strategy for . Let
be the first move of
. For each
, let player
make his/her move by letting
if
is the last move by
. It is clear that whenever
chooses his/her moves in this way, the intersection of the open sets has to be empty.
Suppose that
is a Baire space. Let
be a strategy for the player
. We show that
cannot be a winning strategy for
.
Let be the first move for
. For each open
,
. Apply Lemma 1 to obtain a disjoint collection
consisting of open sets of the form
such that
is dense in
.
For each , we have
for all open
. So the function
is like the function
in Lemma 1. We can then apply Lemma 1 to obtain a disjoint collection
consisting of open sets of the form
such that
is dense in
. Then let
. Based on how
are obtained, it follows that
is dense in
.
Continue the inductive process in the same manner, we can obtain, for each , a disjoint collection
consisting of open sets of the form
(these are moves made by
using the strategy
) such that
is dense in
.
For each , let
. Each
is dense open in
. Since
is a Baire space, every nonempty open subset of
is of second category in
(including
). Thus
. From this nonempty intersection, we can extract a play of the game such that the open sets in this play of the game have one point in common (i.e. player
wins). We can extract the play of the game because the collection
are disjoint. Thus the strategy
is not a winning strategy for
. This completes the proof of Theorem 1.
Theorem 2
The space is of second category in itself if and only if player
has no winning strategy in the game
.
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Some Completeness Properties
Theorem 1 shows that a Baire space is one in which the player has no winning strategy in the Banach-Mazur game (the version in which
makes the first move). In such a space, no matter what strategy player
wants to use, it can be foiled by player
by producing one specific play in which
loses. We now consider spaces in which player
has a winning strategy. A space
is said to be a weakly
-favorable if player
has a winning strategy in the game
. If
always wins, then
has no winning strategy. Thus the property of being a weakly
-favorable space is stronger than the Baire property.
In any complete metric space, the player always has a winning strategy. The same idea used in proving the Baire category theorem can be used to establish this fact. By playing the game in a complete metric space, player
can ensure a win by making sure that the closure of his/her moves have diameters converge to zero (and the closure of his/her moves are subsets of the previous moves).
Based on Theorem 1, any Baire space is a space in which player of the Banach-Mazur game has no winning strategy. Any Baire space that is not weakly
-favorable is a space in which both players of the Banach-Mazur game have no winning strategy (i.e. the game is undecidable). Any subset of the real line
that is a Bernstein set is such a space. A subset
of the real line is said to be a Bernstein set if
and its complement intersect every uncountable closed subset of the real line. Bernstein sets are discussed here.
Suppose is a strategy for
in the game
. If at each step, the strategy
can provide a move based only on the other player’s last move, it is said to be a stationary strategy. For example, in the partial play
, the strategy
can determine the next move for
by only knowing the last move of
, i.e.,
. A space
is said to be
-favorable if player
has a stationary winning strategy in the game
. Clearly, any
-favorable spaces are weakly
-favorable spaces. However, there are spaces in which player
has a winning strategy in the Banach-Mazur game and yet has no stationary winning strategy (see [2]). Stationary winning strategy for
is also called
-winning tactic (see [1]).
Reference
- Choquet, G., Lectures on analysis, Vol I, Benjamin, New York and Amsterdam, 1969.
- Deb, G., Stategies gagnantes dans certains jeux topologiques, Fund. Math. 126 (1985), 93-105.
- Engelking, R., General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
- Haworth, R. C., McCoy, R. A., Baire Spaces, Dissertations Math., 141 (1977), 1 – 73.
- Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.
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Revised 4/4/2014.