Results about Proximal and Semi-proximal Spaces

Proximal spaces were defined by J. Bell as those topological spaces $X$ with a compatible uniformity $U$ on which Player I has a winning strategy in the so-called proximal on $(X,U)$. Nyikos defined the class of semi-proximal spaces where Player II has no winning strategy on $(X,U)$ with respect to some compatible uniformity. The primary focus of this thesis is to study the relationship between the classes of semi-proximal spaces and normal spaces. Nyikos asked whether semi-proximal spaces are always normal. The main result of this thesis is the construction of two counterexamples to this question. We also examine the characterization of normality in subspaces of products of ordinals, relating it to the class of semi-proximal spaces in finite power of $\omega 1$. In addition, we introduce a strengthening of these classes by restricting the proximal game to totally bounded uniformities. We study connections between the proximal game, the Galvin game, and the Gruenhage game. Further, we explore the relationship between semi-proximality and other convergence properties.