Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods

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Theorem

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

Let $V_1$ and $V_2$ be compact sets in $T$.


Then $V_1$ and $V_2$ have disjoint neighborhoods.


Lemma

Let $(X, \tau)$ be a Hausdorff space.

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

Let $x \in X \setminus C$.


Then there are open sets $U$ and $V$ such that $x \in U$, $C \subseteq V$, and $U \cap V = \varnothing$.


Proof

Let $\mathcal F$ be the set of all ordered pairs $\left({Z, W}\right)$ such that:

$Z, W \in \tau$
$V_1 \subseteq Z$
$Z \cap W = \varnothing$

By the lemma, $\operatorname{Im}\mathcal F$ covers $V_2$.

By the definition of compact space, there is a finite subset $K$ of $\operatorname{Im} \mathcal F$ which also covers $V_2$.

By the definition of topology, $\bigcup K$ is open.

By the Principle of Finite Choice, there is a bijection $\mathcal G \subseteq \mathcal F$ such that $\operatorname{img} \mathcal G = K$.


Then $\mathcal G$, and hence its preimage, will be finite.

Let $J = \bigcap \operatorname{Im}^{-1} \mathcal G$

By Subset of Intersection, $V_1 \subseteq J$.

By the definition of a topology, $J$ is open.

Then $\bigcup K$ and $J$ are disjoint open sets such that $V_2 \subseteq \bigcup K$ and $V_1 \subseteq J$.

$\blacksquare$


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