Cloz-Covers of Tychonoff Spaces

Abstract

Abstract. In this paper, we construct a cover (L(X), c X ) of a space X such that for any cloz-coverUsing this, we show that every Tychonoff space X has a minimal cloz-cover (E cc (X), z X ) and that for a strongly zero-dimensional space X, βE cc (X) = E cc (βX) if and only if Ecc(X) is z # -embedded in Ecc(βX).

Wallman Covers and Stone-Čech Compactifications of Cloz Covers

“…proved similar results for the category of Hausdorff spaces and perfect continuous maps (the category of regular spaces and perfect continuous maps, resp.). To generalize extremally disconnected spaces, basically disconnected spaces, quasi-F spaces and cloz-spaces have been introduced and their minimal covers have been studied by various authors [3], [6], [7], [8], [10], [13] .…”

“…In [10], it is shown that for a weakly Lindelöf space X, βE cc (X) = E cc (βX) and every spaces has a minimal cloz-cover.…”

Minimal Cloz-Covers and Boolean Algebras

“…Then βE cc (X) = E cc (βX) if and only if E cc (X) is z # -embedded in E cc (βX), that is, for any A ∈ Z(E cc (X)) # , there is a B ∈ Z(E cc (βX)) # such that A = B∩E cc (X). Morever, if βE cc (X) = E cc (βX), then z X : E cc (X) −→ X is z # -irreducible, that is, z X (Z(E cc (X)) # ) ⊆ Z(X) # ( [7]). For any strongly zero-dimensional space X, S(G(βX)) and βX are zero-dimensional space and hence E cc (βX) is a zero-dimensional space.…”

“…Then E cc (X) is the subspace [6]). Moreover, it was shown that every space has a minimal cloz-cover( [7]). …”

“…Open questions in the theory of clozspaces concerns with the minimal cloz-covers of non-compact spaces and the relation between E cc (βX) and E cc (X)( [3]). For this problem, we have partial answers in [6] and [7]. Indeed, it is shown that for a weakly Lindelöf space X, βE cc (X) = E cc (βX)( [4], [6]) and every spaces has a minimal cloz-cover( [7]).…”