Subcover is Refinement of Cover

Theorem
Let $S$ be a set.

Let $\mathcal C$ be a cover for $S$.

Let $\mathcal V$ be a subcover of $\mathcal C$.

Then $\mathcal V$ is a refinement of $\mathcal C$.

Corollary
Let $T = \left({X, \vartheta}\right)$ be a topological space.

Let $\mathcal U$ be an open cover for $S$.

Let $\mathcal V$ be a subcover of $\mathcal U$.

Then $\mathcal V$ is an open refinement of $\mathcal U$.

Proof
From the definition of subcover:
 * $\mathcal V \subseteq \mathcal C$

That is, every element of $\mathcal V$ is an element of $\mathcal C$.

From the definition of subset, every element of $\mathcal V$ is the subset of some element of $\mathcal C$.

This is precisely the definition of refinement.

Proof of Corollary
As all the elements of $\mathcal U$ are open, all the elements of $\mathcal V$ are likewise open.

Hence the result from definition of open refinement.