Infinite Particular Point Space is not Countably Metacompact

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Theorem

Let $T = \left({S, \tau_p}\right)$ be an infinite particular point space.


Then $T$ is not countably metacompact.


Proof

Suppose $T$ is a countable particular point space.

Let $\mathcal C$ be the open cover of $T$ defined as:

$\mathcal C =\left\{{\left\{{x, p}\right\}: x \in S}\right\}$

$\mathcal C$ is countable and has no open refinement except $\mathcal C$ itself.

But $\mathcal C$ is not point finite because $\forall U \in \mathcal C: p \in U$, and $\mathcal C$ is (countably) infinite.


Now suppose $T$ is an uncountable particular point space.

Let $S' \subseteq S$ be a countable subset of $S$.


Let $\mathcal C$ be the open cover of $T$ defined as:

$\mathcal C = \left\{{\left\{{x, p}\right\}: x \in S'}\right\} \cup \left\{{S \setminus S' \cup \left\{{p}\right\}}\right\}$


$\mathcal C$ is also countable and has no open refinement except $\mathcal C$ itself.


And similarly, $\mathcal C$ is not point finite because $\forall U \in \mathcal C: p \in U$, and $\mathcal C$ is (countably) infinite.


Hence the result by definition of countably metacompact.

$\blacksquare$


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