Fixed Point Set of Continuous Self-Map on Hausdorff Space is Closed

Theorem
Let $T = \struct {S, \tau}$ be a Hausdorff space.

Let $f: T \to T$ be a continuous mapping on $T$.

Let $W$ be the set defined as:
 * $W = \set {x \in T: \map f x = x}$

Then $W$ is closed in $T$.

Proof
Let $g: T \to T$ be the identity mapping on $T$:
 * $\forall x \in T: \map g x = x$

From Identity Mapping is Continuous, $g$ is a continuous mapping on $T$.

From Equal Images of Mappings to Hausdorff Space form Closed Set:
 * $\set {x \in T: \map f x = \map g x}$ is closed in $T$.

and the result follows.