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