Speaker: Dr. Steven Clontz, from the University of South Alabama
Date and Time: Friday, January 21, 2022, 12:00-1:00 via Zoom. Please contact Will Brian to obtain the Zoom link.
Title: Games Topologists Play
Abstract: Several ideas from topology and set theory may be characterized by considering two-player infinite-length games. During each round n ∈ {0,1,2, . . . }, suppose Player 1 makes a move an (perhaps choosing an open cover of a given regular space), followed by Player 2 making a move bn (perhaps choosing a finite subcollection from 1’s chosen cover); the winner of such a game is determined by the sequence of moves ( a0,b0,a1,b1, . . . ) (perhaps Player 2 wins if their choices form a cover).
The topological game specified above is known as Menger’s game, and Player 2 has an unbeatable strategy that only uses information limited to the round number and the most recent move of Player 1 in this game if and only if the given regular space is σ-compact. In this talk, we will explore various results of this flavor found in the literature, including an interesting game-theoretic proof appropriate for undergraduates that the real numbers are uncountable.