Point in Finite Hausdorff Space is Isolated/Proof 2
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Theorem
Let $T = \struct {S, \tau}$ 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 $T = \struct {S, \tau}$ be a $T_2$ (Hausdorff) space.
Let $X \subseteq T$ be finite.
From Separation Properties Preserved in Subspace, it follows that $\struct {X, \tau_X}$ is also a $T_2$ (Hausdorff) space.
From $T_2$ Space is $T_1$ Space it follows that $\struct {X, \tau_X}$ is a $T_1$ (Fréchet) space.
From Finite $T_1$ Space is Discrete, it follows that $\struct {X, \tau_X}$ is a discrete space.
The result follows from Topological Space is Discrete iff All Points are Isolated.
$\blacksquare$