Identity Mapping on Metric Space is Homeomorphism

Theorem
Let $M = \left({A, d}\right)$ be a metric space.

The identity mapping $I_A: M \to M$ defined as:
 * $\forall x \in A: I_A \left({x}\right) = x$

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.