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.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $4$: The Hausdorff condition: Exercise $4.3: 6$