Point in Finite Hausdorff Space is Isolated/Proof 2

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Theorem

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

Let $X \subseteq S$ such that $X$ is finite.

Let $x \in X$.


Then $x$ is isolated in $X$.


Proof

Let $T = \left({S, \tau}\right)$ be a $T_2$ (Hausdorff) space.

Let $X \subseteq T$ be finite.

From Separation Properties Preserved in Subspace, it follows that $\left({X, \tau_X}\right)$ is also a $T_2$ (Hausdorff) space.

From $T_2$ Space is $T_1$ Space it follows that $\left({X, \tau_X}\right)$ is a $T_1$ (Fréchet) space.

From Finite $T_1$ Space is Discrete, it follows that $\left({X, \tau_X}\right)$ is a discrete space.

The result follows from Topological Space is Discrete iff All Points are Isolated.

$\blacksquare$