Metacompactness

Subjects: Topology
Links: Paracompacteness, Collectionwise Normal Spaces, Special Types of Collections in Topology

Def: A topological space is called metacompact if every open cover of $X$ has a point-finite open refinement.

Def: A cover $\{A_{\alpha }\mid \alpha <\kappa \}$ of a space $X$ will be called irreducible if $\bigcup _{\alpha \in I}{A}_{\alpha }\ne X$ for every $I\subset \kappa$.

Prop: Every point-finite cover $\{A_{\alpha }\mid \alpha <\kappa \}$ of a space $X$ has an irreducible subcover.

Obs: Every paracompact space is metacompact.

Th: Every countably compact metacompact space is compact.

Michael-Nagami Theorem: Every metacompact collectionwise $T_4$ space is paracompact.

Lemma: For every open cover $\{U_{\alpha }\mid \alpha <\kappa \}$ of a metacompact space there is a point finite cover $\{V_{\alpha }\mid \alpha <\kappa \}$ of $X$ such that $V_{\alpha }\subseteq U_{\alpha }$ for every $\alpha <\kappa$.

Lemma: If there is a closed, continuous and surjective function $f:X\to Y$, and $X$ is a metacompact space, then every open $\sigma$-point finite cover of $Y$ have a point finite open refinement.

Worrell's Theorem: Let $X$ and $Y$ be $T_2$ spaces. If $f:X\to Y$ be a closed, continuous and surjective function, and $X$ is metacompact, then $Y$ is metacompact.

Th: Let $Y$ be a metacompact space. If $f:X\to Y$ is a perfect function, then $X$ is also metacompact.

Prop: If $X$ is $T_2$ metacompact and $Y$ is $T_2$ and compact, then $X\times Y$ is metacompact.