On the companion of spaces having dense, relatively countable compact subspaces

Abstract

The notion of "pseudocompactness" was introduced by Hewitt. The concept of relatively countably compact subspaces were explored by Marjanovic to show that a $\Psi$-space is pseudocompact. A topological space is said to be DRC (DRS) iff it possesses a dense, relatively countably compact (or relatively sequentially compact, respectively) subspace. The concept of selectively pseudocompact game Sp(X) and the selectively sequentially pseudocompact game Ssp(X) were introduced by Dorantes-Aldama and Shakhmatov. They explored the relationship between the existence of a winning strategy and a stationary winning strategy for player P in these games. In particular, they observed that there exists a stationary winning strategy in the game Sp(X) (Ssp(X)) for Player P iff $X$ is DRC (or DRS, respectively). In this paper we introduce natural weakening of the properties DRC and DRS: a space $X$ is DRCo ( DRSo) iff there is a sequence $(D_n:n \in { \omega})$ of dense subsets of $X$ such that every sequence $(d_n:n \in { \omega} )$ with $d_n \in D_n$ has an accumulation point (or contains a convergent subsequence, respectively). These properties are also equivalent to the existence of some limited knowledge winning strategy on the corresponding games $Sp(X)$ and $Ssp(X)$. Clearly, DRS implies DRC and DRSo, DRC or DRSo imply DRCo. The main part of this paper is devoted to prove that apart from these trivial implications, consistently there are no other implications between these properties.