The impact of infinite games on topological cardinal invariants and function spaces

This thesis is a collection of results in the field of set-theoretic topology, with anemphasis on cardinal invariants and infinite topological games. Our aim is to generalize, strengthen and refine existing theorems and cardinal inequalities as well as answer open questions in this area of knowledge. Chapter 3 presents some game-theoretic improvements on well established cardinal inequalities by Arhangel’skii, Hajnal and Juhász through the use of topological games and elementary submodels. Whereas answers to a question of Tkachuk and one of Clontz and Holshouser about the existence of some winning strategies on the Cp-space of topological spacesis provided and discussed in Chapter 4 as well as analogous results involving similargames.In Chapter 3, we discuss the joint paper with Angelo Bella and Santi Spadarotitled "Cardinal inequalities involving the weak Rothberger and cellularity games"(Bella, Chiozini, and Spadaro, 2025). We study the use of infinite games in the context of cardinal inequalities derived and inspired by three famous such inequalities from the late 1960’s and early 1970’s by Arhangel’skii, Hajnal and Juhász along with their progressive improvements through the years until recently. Specifically, we present a bound to the cardinality of Urysohn and regular spaces as well as a bound to the weak Lindelöf degree of the Gδ-modification of a space using the weak Rothberger game and the cellularity game. We also provide examples to both show thatsuch bounds are strict improvements to previous known ones and to illustrate their sharpness.In Chapter 4, we present the paper "On the connection between games on a spaceand on its Cp-space" (Preprint). We give a negative answer to Tkachuk’s questionwhether it is possible for Player II to have a winning strategy in the game CD overCp(X) provided that the space Cp(X) has the Fréchet-Urysohn property as well asa positive one to Clontz and Holshouser’s question whether it is possible to characterize the Rothberger property (and thus the Rothberger game) on a space X by a point-picking game on Cp(X) by providing something a little stronger: a gameplayed on Cp(X) that is equivalent to the Rothberger game on X. We then present a few more games played on Cp(X) that are equivalent to games played on X.In Chapter 5, we discuss the article "Function spaces and the highly divergentgame" (Preprint), a joint work with Angelo Bella and Santi Spadaro. We present thehighly divergent game inspired by the selectively highly divergent property definedby Jiménez-Flores, Ríos-Herrejón, Rojas-Sánchez, and Tovar-Acosta. We provide a few results concerning when player II has a winning strategy in the highly divergent game as well as sufficient properties on a space that imply that II has a winning strategy in its Cp-space.Finally, in Chapter 6, we discuss part of the joint paper with Tamás Csernák and Lajos Soukup tiled "The point separating game on topological spaces" (Preprint). After defining the point-separating game and the set membership game, we define two correlated cardinal functions: the point separation number and the set membership number, and proceed to provide their evaluation in powers of the Alexandroff Double Arrow space and find bounds to them on product spaces and quasi-developable spaces, a wide class of spaces that includes, e.g., metric spaces.