Compact Subspace of Hausdorff Space is Closed/Proof 2

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Theorem

Let $H = \left({A, \tau}\right)$ be a Hausdorff space.

Let $C$ be a compact subspace of $H$.


Then $C$ is closed in $H$.


Proof

For a subset $S \subseteq A$, let $S^{\complement}$ denote the relative complement of $S$ in $A$.


Consider an arbitrary point $x \in C^{\complement}$.

Define the set:

$\displaystyle \mathcal O = \left\{{V \in \tau: \exists U \in \tau: x \in U \subseteq V^{\complement}}\right\}$

By Empty Intersection iff Subset of Complement, we have that:

$U \subseteq V^{\complement} \iff U \cap V = \varnothing$

Hence, by the definition of a Hausdorff space, it follows that $\mathcal O$ is an open cover for $C$.


By the definition of a compact subspace, there exists a finite subcover $\mathcal F$ of $\mathcal O$ for $C$.

By the Principle of Finite Choice, there exists an $\mathcal F$-indexed family $\left\langle{U_V}\right\rangle_{V \in \mathcal F}$ of elements of $\tau$ such that:

$\forall V \in \mathcal F: x \in U_V \subseteq V^{\complement}$

Define:

$\displaystyle U = \bigcap_{V \mathop \in \mathcal F} U_V$

By General Intersection Property of Topological Space, it follows that $U \in \tau$.


Clearly, $x \in U$.

We have that:

\(\displaystyle U\) \(\subseteq\) \(\displaystyle \bigcap_{V \mathop \in \mathcal F} \left({ V^{\complement} }\right)\) Set Intersection Preserves Subsets
\(\displaystyle \) \(=\) \(\displaystyle \left({\bigcup \mathcal F}\right)^{\complement}\) De Morgan's laws
\(\displaystyle \) \(\subseteq\) \(\displaystyle C^{\complement}\) Definition of Cover of Set and Relative Complement inverts Subsets

From Subset Relation is Transitive, we have that $U \subseteq C^{\complement}$.


Hence $C^{\complement}$ is a neighborhood of $x$.

From Set is Open iff Neighborhood of all its Points, we have that $C^{\complement} \in \tau$.

That is, $C$ is closed in $H$.

$\blacksquare$