Round subsets of Wallman-type compactifications

Abstract

Let ℒ be a normal base of a Tychonoff space X and ωℒ denote the Wallman-type (real-) compactification of X generated by ℒ. This Wallman-type compactification is known to associate with a unique proximity δ. A ℒ-filter ℒ is round if for each F ∈ ℒ there is an Fo ∈ ℒ there is an Fo ∈ ℒ such that Fo(X-F). A subset A of ω(£) is called a round subset of ω (£) iff for each Z ∈ ℒ, if C1w(x)Z contains A, then it is a neighborhood of A. Properties of round ℒ-filters and round sets of ω(ℒ) are introduced. We also prove that the intersection of all the free ℒ-ultrafilters is ℒ= {Z ∈ ℒ: C1x(X-Z) is compact} iff ω(ℒ) – X is a round subset of ω(ℒ) if ℒ is a separating nest generated intersection ring with property (α) then ω(ℒ) - v(ℒ) is a round subset of ω(ℒ).