On nearly paracompact spaces via regular even covers. (English) Zbl 0942.54018

Mukherjee, M. N.; Debray, Atasi

Mat. Vesn. 50, No. 1-2, 23-29 (1998).

Recall that a space \(X\) is nearly paracompact if every regular open cover of \(X\) has a locally finite open refinement. The authors introduce the notion of regular even cover: a cover \(\mathcal U\) of a space \(X\) is called regular even if there is a neighborhood \(V\) of the diagonal \(\Delta_X \subset X\times X\) such that \(V\) is the union of regular open sets and \(\{V[x]:x \in X\}\) is a refinement of \(\mathcal U\). Among nine equivalent assertions on an almost regular space \(X\) there is the following result: \(X\) is nearly paracompact iff every regular open cover of \(X\) is regular even.

Reviewer: Ljubiša Kočinac (Aleksandrovać)

MSC:

54D20

Noncompact covering properties (paracompact, Lindelöf, etc.)

Keywords:

almost regular space; regular even cover