Banach and Mazur game

I want to rewrite here a part of my degree thesis speaking about a very nice way to describe residual sets.. I start with the abstract Bourbakist definition anyway.
Let E be a topological space.

A set S\subset E is called nowhere dense if it fails to be dense in every open subset of E: that is, for every nonempty open U\subset E there is a nonempty open V\subset U\setminus S.

S is then nowhere dense if and only if its closure \bar{S} has empty interior; a finite union of nowhere dense sets is also nowhere dense; the topological boundary of a closed set or of an open set is nowhere dense.


A set A\subset E is called of first (Baire) category if A is a countable union of nowhere dense sets.
A set that is not of first category is called of second category.
The complement of a first category set is called residual.

Residual sets are then sets that are countable intersections of sets with dense interior. If a set is first category, then all its subsets are. In some sense the sets of first category are “small” (they are also called meagre sets because of this) and the sets of second category are “large”. We will show this using a description introduced by Banach and Mazur.

The Banach-Mazur game
Let X be a metric space.

Suppose that there is given a subclass \mathcal{E} of the subsets of X having nonempty interior, such that each open set is contained in some member of \mathcal{E}. Suppose that there are given two sets A\subset X and B=X\setminus A. The game [A,B] is played according to the following rules: two players (A) and (B) alternately choose sets
U_1\supset V_1\supset\ldots\supset U_n\supset V_n\supset\ldots
from the class \mathcal{E}, with (A) choosing the U_i‘s and (B) choosing the V_i‘s, numerably many times. Such a nested sequence is called a play of the game. The player (B) is declared winner if
\bigcap_{i=1}^{\infty}V_i\subset B,
while otherwise (that is, if \bigcap_{i=1}^{\infty}V_i\cap A\neq\emptyset) the winner is (A).

To be more precise, we will call a strategy for (B) a sequence of functions \beta=\{\beta_n\} giving for any sequence (U_1,V_1,\ldots , U_n) of members of \mathcal{E} as above, a new member

V=\beta_n(U_1,V_1,\ldots, V_{n-1},U_n)\in\mathcal{E},\; V\subset U_n

A play of the game (U_i,V_i)_{i=1}^{\infty} is called consistent with a strategy \beta if for all n,
V_n= \beta_n(U_1,V_1,\ldots, V_{n-1},U_n)
A strategy \beta is winning for (B) if every play of the game consistent with \beta ends up with a victory of B. If such a strategy exists, the game is said to be determined in favour of (B).
Clearly, in order to win, (A) will hope that the set A is large, while (B) will have the same hopes for the set B. The right meaning for the word “large” in this context was (conjectured by Mazur and) proved by Banach to be the same as “residual”:

Theorem (Banach-Mazur)
The game [A,B] is determined in favour of player (B) if and only if the set B is residual in X.

“if” part: Let B\supset\cap_{i=1}^{\infty}G_i with G_i open dense sets. We then choose
V_n:=\beta_n(U_1,\ldots, U_n)\subset U_n\cap G_n,
which can be done for the properties of \mathcal{E} and of G_n. The resulting strategy \beta=\{\beta_n\} is then easily seen to be winning, by the definition of the G_i‘s.
“only if” part: Let \beta=\{\beta_n\} be a winning strategy for (B). We will call a consistent sequence of n members of \mathcal{E}  a \beta-chain of order $latex n$, and the interior of $latex V_n$ will be called the interior of the chain. We will now inductively construct a sequence of dense open sets \{G_n\} whose intersection is contained in the set B.
Let F_1 be a maximal subfamily of the \beta-chains of order 1 whose members have disjoint interiors, and let G_1 be the union of the interiors of the members of F_1. This open set is dense by maximality.
Then, among all the \beta-chains of order 2 that are continuations of some member of F_1 we choose a maximal subfamily F_2 with disjoint interiors, and let G_2 be the union of these interiors. The open set G_2 is again dense by maximality.
Iterating this procedure, we get a sequence of dense open sets set{G_n}. If x\in \cap_{i=1}^{\infty}G_i then there is a unique sequence \{C_n\} of \beta-chains C_n\in F_n such that for each n, x is in the interior of C_n. These \beta-chains are ordered by continuation and the limit play is consistent with \beta, so it must be a winning one. Thus x\in B, and since x was arbitrary, B\supset\cap_i G_i.\square

As an application, we cite the following, which is usually proved using directly the definitions. We give instead a proof using the Banach-Mazur game:


If the metric space X is complete and B\subset X is a dense G_{\delta} set (i.e. a countable intersection of open sets), then B is residual.

Let B=\cap_m A_m, where the sets A_m are open. We take \mathcal{E} be the family of all closed balls, and we describe the winning strategy of player (B) in the game [X\setminus B, B]: let U_n be the closed ball chosen by (A) at his n-th move. Since B is dense, B\cap intU_n is nonempty. So intU_n\cap A_n is a nonempty open set. By regularity there exists a ball of radius less than 2^{-n}  whose closure is contained in the above set. We then choose that ball as V_n. We obtain that the sequence (U_{i+1}) refines the Cauchy filter basis (V_i) and by completeness it has as intersection a single point x, contained in each A_n, and thus in B. \square

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