Identity Mapping is Homeomorphism
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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.
$\blacksquare$