Identity Mapping is Continuous

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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.

Metric Space

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

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

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

is a continuous mapping.


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