Between Countable Compactness and Pseudocompactness: Dense Subspaces, Selection Principles, and Topological Games

The classical implication: countably compact implies pseudocompact cannot, in general, be reversed. In this talk we investigate several refinements and weakenings of countable compactness that lie between countable compactness and pseudocompactness, with special emphasis on selection principles and their game-theoretic counterparts. 

We first discuss first countable spaces that are pseudocompact but not countably compact}, and present constructions, in particular under {CH} or s=c. 
Next, we turn to selective versions of pseudocompactness. A space is selectively pseudocompact if from every sequence of pairwise disjoint non-empty open sets one can select points whose sequence has an accumulation point. We introduce and compare several natural variants of this selection principle, show that they are genuinely distinct, and give a consistent negative answer to the question whether regular pseudocompact spaces of countable tightness must be selectively pseudocompact.

Finally, we describe the connection between these compactness-type properties and open-point topological games. We show that the existence of stationary or Markov winning strategies for Player P in the games Sp(X) and Ssp(X) corresponds precisely to dense (relatively) compactness-type properties and their natural weakenings.Apart from the obvious implications, we prove that no further implications hold between the resulting hierarchy of properties.

The talk highlights how selection principles and topological games together provide a refined structural picture between countable compactness and pseudocompactness, particularly in the realm of first countable spaces.