Separable Metacompact Space is Lindelöf

Theorem
Let $T = \left({X, \vartheta}\right)$ be a separable topological space which is also metacompact.

Then $T$ is a Lindelöf space.

Proof
$T$ is separable iff there exists a countable subset of $X$ which is everywhere dense.

$T$ is metacompact iff every open cover of $X$ has an open refinement which is point finite.

$T$ is a Lindelöf space if every open cover of $X$ has a countable subcover.

Having established the definitions, we proceed.

Let $\mathcal U$ be an open cover of $X$ which has no countable subcover.

Let $\mathcal V$ be an open refinement which is point finite but uncountable.

Let $\mathcal S$ be a countable subset of $X$ which is everywhere dense.

Then each $V \in \mathcal V$ contains some $s \in \mathcal S$.

So some $s \in \mathcal S$ is contained in an uncountable number of elements of $\mathcal V$.

This contradicts the nature of $\mathcal V$.