# Real Numbers and Infinite Games, Part II

In my last post, I wrote about two infinite games whose analysis leads to interesting questions about subsets of the real numbers.  In this post, I will talk about two more infinite games, one related to the Baire Category Theorem and one to Diophantine approximation.  I’ll then talk about the role which such Diophantine approximation questions play in the theory of dynamical systems.

The Choquet game and the Baire Category Theorem

The Cantor game from Part I of this post can be used to prove that every perfect subset of ${\mathbf R}$ is uncountable.  There is a similar game which can be used to prove the Baire Category Theorem.  Let $X$ be a metric space.   In Choquet’s game, Alice moves first by choosing a non-empty open set $U_1$ in $X$.  Then Bob moves by choosing a non-empty open set $V_1 \subseteq U_1$.  Alice then chooses a non-empty open set $U_2 \subseteq V_1$, and so on, yielding two decreasing sequences $U_n$ and $V_n$ of non-empty open sets with $U_n \supseteq V_n \supseteq U_{n+1}$ for all $n$.  Note that $\bigcap U_n = \bigcap V_n$; we denote this set by $U$.  Alice wins if $U$ is empty, and Bob wins if $U$ is non-empty. Continue reading