Identity Mapping is Homeomorphism

Theorem

Let $T$ be a topological space.

The identity mapping $I_T: T \to T$ defined as:

$\forall x \in T: \map {I_T} x = x$

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.

$\blacksquare$