A Compactification with θ-Continuous Lifting Property

Abstract

1. Let X be a topological space, and let X′ be the set of all non-convergent ultrafilters on X. If A ⊆ X, let , and A* = A ∪ A′. If is a filter on X such that for all , then let. be the filter on X* generated by ; let be the filter on X* generated by . If exists then ; otherwise, .A convergence is defined on X* as follows: If x ∈ X, then a filter A → x in X* if and only if , where Vx(x) is the X neighborhood filter at x; , then in X* if and only if .

A Compactification for Convergence Ordered Spaces

“…If X has the trivial order relation, then (oX*, 4>) is the topological compactification described in [2]. Note that when oX* is T 2 , this compactification coincides with the topological Stone Cech compactification.…”

“…If X is a convergence ordered space, then X is said to be TVordered if i(x) and d(x) are closed sets for each x G X. If x < y whenever g -> x, © -> y, and g^ ©, then X is defined to be T 2 (Si) If g->x, ©GX', and g<®, then z/(©) ç x. (S 2 ) Kg-^x, @GX', and @<g, then *(®)çx.. a 7\ (respectively, T 2 ) c.o.s.…”

“…A topological ordered compactification. In [2], we showed that the convergence space compactification of [4] gives rise to a topological compactification with interesting properties. A similar procedure is used here to construct a topological ordered compactification of an arbitrary topological ordered space.…”

“…Note that when oX* is T 2 , this compactification coincides with the topological Stone Cech compactification. However oX* is rarely T 2 ; for circumstances under which this occurs, see Theorem 2.5, [2].…”