Point in Finite Hausdorff Space is Isolated/Proof 2

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.

$\blacksquare$