Editing Meagre set (section)

==Banach–Mazur game==

Meagre sets have a useful alternative characterization in terms of the [[Banach–Mazur game]]. 
Let <math>Y</math> be a topological space, <math>\mathcal{W}</math> be a family of subsets of <math>Y</math> that have nonempty interiors such that every nonempty open set has a subset belonging to <math>\mathcal{W},</math> and <math>X</math> be any subset of <math>Y.</math> 
Then there is a Banach–Mazur game <math>MZ(X, Y, \mathcal{W}).</math> 
In the Banach–Mazur game, two players, <math>P</math> and <math>Q,</math> alternately choose successively smaller elements of <math>\mathcal{W}</math> to produce a sequence <math>W_1 \supseteq W_2 \supseteq W_3 \supseteq \cdots.</math> 
Player <math>P</math> wins if the intersection of this sequence contains a point in <math>X</math>; otherwise, player <math>Q</math> wins. 

{{math theorem|name=Theorem|note=|style=|math_statement=
For any <math>\mathcal{W}</math> meeting the above criteria, player <math>Q</math> has a [[winning strategy]] if and only if <math>X</math> is meagre.
}}