Identity Mapping is Continuous

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

The identity mapping $I_S: S \to S$ defined as:
 * $\forall x \in S: \map {I_S} x = x$

is a continuous mapping.

Proof
Let $U \in \tau$.

We have Identity Mapping is Bijection.

So $I_S^{-1}$ is well-defined and:
 * $\forall x \in U: \map {I_S^{-1} } x = x$

Thus $I_S^{-1} \sqbrk U = U \in \tau$.

Hence, by definition of continuous mapping, $I_S$ is continuous.

Also see

 * Definition:Finer Topology
 * Definition:Coarser Topology


 * Identity Mapping to Coarser Topology is Continuous