(a)-spaces and selectively (a)-spaces from almost disjoint families

Selectively (a)-spaces from almost disjoint families are necessarily countable under a certain parametrized weak diamond principle

“…In [20], the second author has investigated the presence of the selective version of property (a) in a certain class of topological spaces, the Mrówka-Isbell spaces from almost disjoint families, and, as expected, many aspects of such presence within this class have combinatorial characterizations or, at least, are closely related to combinatorial and set-theoretical hypotheses. Let us recall how such spaces are constructed.…”

“…The following theorem is a general result on selectively (a)-spaces (not only for those from almost disjoint families) and was established by the second author in [20]. Recall that the density of a topological space X, d(X), is the minimum of the cardinalities of all dense subsets of X, provided this is an infinite cardinal, or is ω = ℵ 0 otherwise.…”

“…However, it is also shown in [20] that 2 ℵ0 < 2 ℵ1 alone does not avoid the existence of uncountable selectively (a) almost disjoint families. Our main goal in this paper is to show that a certain effective parametrized weak diamond principle is enough to ensure countability of the almost disjoint family in this context.…”

On star covering properties related to countable compactness and pseudocompactness

“…The reader may find a collection of results on (a)-spaces and selectively (a)spaces in [12], [9], [3] and [21]. It is worthwhile remarking that consistent combinatorial hypotheses provide the consistency of the non-existence of uncountable selectively (a)-spaces from almost disjoint families (see [18] and [15]; for related results on separable, locally compact selectively (a)-spaces in general, see [19]). A survey of recent results on star selection principles may be found in [10].…”

“…It is very usual, in the star covering properties literature, the research programme of looking for conditions (either in a topological space or in a combinatorial structure) under which, after carrying out a specific construction, we get examples of topological spaces satisfying a given topological star covering property (or, more recently, a star selection principle). In the case of almost disjoint families, Szeptycki and Vaughan investigated and characterized the almost disjoint families for which the corresponding Ψ-space satisfy property (a) ( [22]), and the third author have did the same for the selective version of property (a) ( [18]).…”