submesocompact

Definition

A topological space X is called submesocompact if for every open cover U of X, there exists a sequence U_n of open covers of X which satisfies:

1. every U_n refines U;
2. for every nonempty compact set K in X, there exists some n such that finitely many member of U_n meets K.

Reference

Shou Lin, Mapping theorems on k-semistratifiable spaces, Tsukuba J. Math. Vol.21 No.3 (1997), 809-815.