Star covering properties

Abstract

A space X is discretely absolutely star-Lindelöf if for every open cover U of X and every dense subset D of X, there exists a countable subset F of D such that F is discrete closed in X and St(F, U) = X, where St(F, U) = {U ∈ U : U ∩F = ∅}. We show that every Hausdorff star-Lindelöf space can be represented in a Hausdorff discretely absolutely star-Lindelöf space as a closed G δ-subspace.

On Starcompactness Versus Countable Pracompactness

“…For regular spaces, (DFCC) is equivalent to 3-starcompactness [17]. Or even better: for regular spaces, (DFCC) is equivalent to 2 1 2 -starcompactness [9]. We also have a well-known theorem.…”

“…We notice that n-starcompact spaces in our terminology are called n-pseudocompact in [4] and strongly n-starcompact in [9], while n 1 2 -starcompact spaces are named n-starcompact in [9].…”

“…Although the study of star covering properties of a topological space could be dated back to the seventies or even earlier [1,2,10,11,15,22], a systematic investigation of them was done first by van Douwen, G. Reed, A. Roscoe and I. Tree [9] in 1991. Since then, this area has generated substantial interest of many topologists [6,7,8,13,14,16,20,21,23].…”

Star-Hurewicz and related properties

“…Recall that a space X is strongly starcompact [3], [13] if for each open cover U of X there is a finite set F ⊂ X such that St(F, U) = X. Clearly, every strongly starcompact space is SSH.…”

“…A number of the results in the literature show that many topological properties can be described and characterized in terms of star covering properties (see [3], [13], [2], [12]). The method of stars has been used to study the problem of metrization of topological spaces, and for definitions of several important classical topological notions.…”