Separable Metacompact Space is Lindelöf/Proof 2

Theorem
Let $T = \left({X, \tau}\right)$ 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 $S$ be a countable dense subset of $X$.

Let $\mathcal U$ be an open cover of $X$.

Let $\mathcal V$ be a point finite open refinement of $\mathcal U$.

By Point-Finite Open Cover of Separable Space is Countable, $\mathcal V$ is countable.

By the Axiom of Countable Choice, there is a mapping $G: \mathcal V \to \mathcal U$ such that:
 * $\forall V \in \mathcal V: V \subseteq G \left({V}\right)$

Then $G \left({\mathcal V}\right)$ is a countable subcover of $\mathcal U$.

Thus each open cover of $X$ has a countable subcover, so $T$ is a Lindelöf space.