TWO OPEN-POINT GAMES RELATED TO SELECTIVE (SEQUENTIAL) PSEUDOCOMPACTNESS, WITH APPLICATION TO 1-CL-STARCOMPACTNESS PROPERTY OF MATVEEV (Research Trends on Set-theoretic and Geometric Topology and their cooperation with various branches)

Abstract

A topological space X is selectively sequentially pseudocompact (selectively pseudocompact) if for every sequence {U_{n} : n in mathrm{N}} of non-empty open subsets of X, one can choose a point x_{n} in U_{n} for every n in mathrm{N} in such a way that the sequence {x_{n} : n in mathrm{N}} has a convergent subsequence (respectively, has an accumulation point in X). It was shown by the authors in [3] that the class of selectively sequentially pseudocompact spaces is closed under taking arbitrary products and continuous images, contains the class of dyadic spaces and forms a proper subclass of the class of selectively pseudocompact spaces. Moreover, the latter class coincides with the class of strongly pseudocompact spaces of García-Ferreira and Ortiz-Castillo [7]. In this paper, we define two topological games closely related to the class of selectively (sequentially) pseudocompact spaces. Let X be a topological space. At round n, Player A chooses a non-empty open subset U_{n} of X, and Player B responds by selecting a point x_{n} in U_{n}. In the selectively sequentially pseudocompact game Ssp(X), Player B wins if the sequence {x_{n} : nin mathrm{N}} has a convergent subsequence; otherwise Player A wins. In the selectively pseudocompact game Sp(X), Player B wins if the sequence {x_{n} : nin mathrm{N}} has an accumulation point in X; otherwise Player A wins. The (non-)existence of winning strategies for each player in the game Ssp(X) (in the game Sp(X)) defines a compactnesslike property of X sandwiched between sequential compactness (countable compactness) and selective sequential pseudocompactness (selective pseudocompactness) of X. We prove that a topological space X such that Player A does not have a winning strategy in Sp(X), is 1-cl-starcompact in the sense of Matveev. As an application of this result, we give an example of a locally compact, first-countable, zero-dimensional, 1-cl-starcompact space without a dense relatively countably compact subspace. This shows that Theorem 15 in Matveev's survey [10] is not reversible.