supersubmetacompact

Definition

A topological space X is said to be supersubmetacompact iff for every open cover U of X, there exist countably many collections G_n of clopen sets which satisfies the following conditions:

1. Each G_n refines U^F, where U^F is the collection of all unions of finite subcollections from U;
2. For point x, there is an n such that x is contained in no more than finitely many members of G_n.

Reference

D. Buhagiar, T. Miwa, and B. A. Pasynkov, Superparacompact type properties, Yokohama Math. J. Vol.46, pp.71-86 (1998).