Partially ordered set of zero-dimensional one-point extensions of a topological space

Abstract

Given a topological space X, there is a natural relation ≤ on the set E∘(X) of all, up to equivalence, zero-dimensional one-point extensions of X which makes it a partially ordered set. We present results on the partially ordered set 〈E∘(X),≤〉. We associate with each zero-dimensional one-point extension of X, a clopen bornology on X which makes it possible to single out some properties of the partially ordered set 〈E∘(X),≤〉. For instance, we characterize the maximal (resp., minimal) elements of E∘(X). We prove that 〈E∘(X),≤〉 is always a lower semi-lattice. For each Y∈E∘(X), the principal up-set ↑Y is shown to be a complete partially ordered set, while the principal down-set ↓Y is a distributive lattice. We show that if X is also locally compact, the partially ordered set E∘(X) determines the topology of the remainder ζX∖X, where ζX is the Banaschewski compactification of X. Finally, we describe the Banaschewski compactification of every zero-dimensional one-point extension of X by means of its associated clopen bornology.