Mazur game

Mazur game

English

Proper noun

Mazur game

1. (topology) A topological game invented by the Polish mathematician Mazur in order to illustrate the difference between sets of the first category and second category.

   In the Mazur game, the unit interval [0,1] is initially divided into two complementary sets, A and B. Player 1 then chooses a closed interval in [0,1] of length ${\\displaystyle \\leq {1 \\over 2}}$ and greater than zero. Player 2 then chooses an interval within the previously chosen interval, of length ${\\displaystyle \\leq {1 \\over 3}}$ and greater than zero. Player 1 then chooses an interval within the previous interval, of length ${\\displaystyle \\leq {1 \\over 4}}$ and greater than zero, and so on. After infinitely many steps, the intersection of all the chosen intervals is a single point: if it falls within set A, then player 1 wins; if it falls within set B, player 2 wins.