Extensions and Absolutes of Hausdorff Spaces

Abstract

Abstract. In this paper we prove the existence of a completion of a T0-pre-uniform space (X, U), with the property that each Cauchy filter in (X, U) contains a weakly round filter.2000 AMS Classification: 54A05, 54A20, 54E15.

Cloz-Covers of Tychonoff Spaces

“…A sublattice of R(X) is a subset of R(X) that contains ∅, X and is closed under finite joins and finite meets( [7]). Let…”

“…Recall that a map f : Y −→ X is called a covering map if it is an onto continuous, perfect, and irreducible map( [7]). …”

“…For any covering map f : Y −→ X and a subspace S of X, the restriction and corestriction f 0 : f −1 (S) −→ S of f with respect to f −1 (S) and S, respectively is a covering map( [7]). Hence we have the following : (⇐) Suppose that X is not a cloz-space.…”

“…Since f and f β are covering maps, f β (βY − Y ) ⊆ βX − X and l is an onto map, because f β is on onto map( [7]). Since z 0 βX • l = f is a covering map and l is an onto map, l is a covering map.…”

“…Since βX is a zero-dimensional space and f is a covering map, βE cc (X) is a zero-dimensinal space( [7]). Hence there is a clopen set B in βE cc (X) such that p ∈ B and q / ∈ B.…”