Round subsets of Wallman-type compactifications

Abstract

Let ℒ be a normal base of a Tychonoff spaceXand ωℒ denote the Wallman-type (real-) compactification ofXgenerated by ℒ. This Wallman-type compactification is known to associate with a unique proximity δ. A ℒ-filter ℒ is round if for eachF∈ ℒ there is anFo∈ ℒ there is anFo∈ ℒ such thatFo(X-F). A subsetAof ω(£) is called a round subset of ω (£) iff for eachZ∈ ℒ, ifC1w(x)Zcontains A, then it is a neighborhood ofA. 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) –Xis 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).