Finite Topological Space is Compact

Theorem
Let $T = \left({S, \vartheta}\right)$ be a topological space where $S$ is a finite set.

Then $T$ is compact.

Proof by selecting finite subcover
Let $\mathcal V$ be an open cover of $T$.

For each $x\in S$, define $\mathcal{V}_x$ to be $\{V \in \mathcal V : x\in V\}$

Since $S$ is finite, and since by definition of a cover, each $x\in S$ is contained in at least one $V$ in $\mathcal{V}$, we have that $\{\mathcal{V}_x : x \in S\}$ is a finite collection of nonempty sets.

Since choice functions exist for finite collections of nonempty sets, there is a choice function which selects one $V_x$ from $\mathcal{V}_x$ for each $x\in S$. By definition of $\mathcal{V}_x$, such a $V_x$ contains $x$.

Since there were only finitely many $\mathcal{V}_x$, this provides finitely many open sets $V_x \in \mathcal V$ such that $\displaystyle S \subseteq \bigcup_{x\in S} V_x$.

Thus $\{V_x : x\in S\}$ is a finite subcover of $\mathcal V$.

The result follows by definition of compact.

Proof by finiteness of the topology
Since $S$ is finite, it's set of all subsets is finite.

Since the topology $\vartheta$ is by definition a collection of subsets of $S$, it follows that $\vartheta$ is finite as well.

Thus, if $\mathcal{V}$ is an open cover of $S$, then by definition it is a subset of $\vartheta$, and consequently it is already a finite subcover.

The result follows by definition of compact.