C-metacompact Spaces II
by
Elise M. Grabner
Slippery Rock University
Coauthors: Gary C. Grabner

Suppose that X is a topological space and C is a subset of X. We say that X is C-metacompact provided every open cover V of X has an open refinement U such that U_x = {U in U : x in U} is finite for all x in C. We continue to investigate basic properties of this class of spaces and other relative topological properties of metacompactness type based on concepts introduced by A. Arkhangel'skii and I. Gordienko.

Theorem A space X and any subset C of X, X is C-metacompact if and only if [`C] is nearly metacompact on C.

Corollary Suppose that C is a subset of the orthocompact space X. Then X is C-metacompact if and only if [`C] is metacompact.

Corollary For a radial space X and a subset C of X, if X is C-metacompact then [`C] is metaLindelöf.

Theorem The perfect preimage Y of an X^o-metacompact space X is irreducible but not necessarily Y^o-metacompact, where X^o and Y^o are the nonisolated points of X and Y.

Date received: February 8, 1998