Locally Compact Spaces with Defects

Abstract

We call a topological space X a locally compact space with defects if all points in X possess compact neighborhoods except for some points. We investigate this weaker version of local compactness. We show that for x ∈ X• if the partition of singletons of X\(X• ∪ (U\U)) is locally finite, where U ̸= X is an open neighborhood of x, then X is a Tychonoff space. Let X be a T1c locally compact space with defects such that each x ∈ X• has an open neighborhood U such that U is a union of pairwise disjoint compact subsets S s∈S Fs. Then, we show that if the family {Fs}s∈S is locally finite except for a finite number of points, then X is a Tychonoff space.