Identity Mapping on Metric Space is Homeomorphism

Theorem

Let $M = \struct {A, d}$ be a metric space.

The identity mapping $I_A: M \to M$ defined as:

$\forall x \in A: \map {I_A} x = x$

is a homeomorphism.

Proof

We have Identity Mapping is Bijection.

We also have Identity Mapping is Continuous.

Hence, by definition, $I_T$ is a homeomorphism.

$\blacksquare$