On properties of relative metacompactness and paracompactness type. (English) Zbl 1026.54016

Grabner, Elise M.; Grabner, Gary C.; Miyazaki, Kazumi

Topol. Proc. 25(Summer), 145-177 (2000).

Summary: We study several natural relative properties of metacompactness and paracompactness types and the relationships among them. Connections to other relative topological properties are also investigated.
Theorem: Suppose \(C\) and \(F\) are subspaces of the \(T_3\) space \(X\). If \(C\) is strongly metacompact in \(X\) and \(F\) is strongly countably compact in \(X\) then \(C\cap F\) is compact in \(X\).
Theorem: A \(T_2\) space \(X\) is compact if and only if it is normal and strongly metacompact in every larger regular space.
Example A Tychonoff space \(X\) having a subset \(C\) which is 2-paracompact in \(X\) but not metacompact in \(C\) from outside.
Theorem: Suppose \(f:X\to Y\) is a closed mapping onto \(Y\) and \(C\subseteq X\). If \(C\) is cp-metacompact in \(X\) then \(f(C)\) is cp-metacompact in \(Y\).

MSC:

54D20	Noncompact covering properties (paracompact, Lindelöf, etc.)
54B05	Subspaces in general topology
54C05	Continuous maps

Keywords:

relative topological property; metacompact; paracompact; nearly metacompact; closed mapping