Topological Properties Defined By Games and Applications
by
Jiling Cao
The University of Auckland, New Zealand
Coauthors: Warren Moors, Ivan Reilly

Let X be a topological space, and let F be a filter in X. In this talk, we will consider the following G(F)-game played in X between two players A and B. Player A goes first and chooses a point x₁ in X, then player B respond by choosing a member F₁ in F. Following this, player A selects another point x₂ in F₁, and in turn player B again responds by choosing a member F₂ in F. Repeating this process infinitely, the players A and B produce a play of the G(F)-game. We call that B wins a play if the sequence (x_n: n in N) has a cluster point. Otherwise, A is said to have won this play. The filter is called a \sigma-filter if the player B has a wining strategy \sigma for the G(F)-game played in X. Furthermore, the space X is said to have property ( * * ) if every \sigma-filter in X has a cluster point. We will investigate topological spaces which possess property ( * * ). In particular, we show certain types of generalized metric spaces have property ( * * ). Consequently, all Dieudonne-complete spaces, all paracompact spaces and all stratifiable spaces have such property. Finally, we apply property ( * * ) to study active boundaries and kernels of multifunctions. As a result, the Choquet-Dolecki theorem and the classical Vainstein lemma are generalized.

Date received: May 15, 1998