The Banach-Mazur game is an infinite two-player game that characterizes sets of first category. In a
version on the real line, fix a target set 
. Players 
and 
alternately choose nondegenerate closed
intervals 
, 
, 
, with each interval
contained in the interior of the preceding interval
and with lengths tending to zero. The intersection
consists of a single point 
. Player 
wins if 
, and player 
wins otherwise.
With this convention, 
has a winning strategy iff
the complement of 
is of first category in
some nonempty open interval. Player 
has a winning strategy iff 
is of first
category. Mazur's original nested-interval game
is Problem 43 of the Scottish Book Problems
(Mauldin 2015, pp. 116-117).
