Identity Mapping is Continuous

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Theorem

Let $T = \left({S, \tau}\right)$ be a topological space.


The identity mapping $I_S: S \to S$ defined as:

$\forall x \in S: I_S \left({x}\right) = x$

is a continuous mapping.


Metric Space

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


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

$\forall x \in A: I_A \left({x}\right) = x$

is a continuous mapping.


Proof

Let $U \in \tau$.

We have Identity Mapping is Bijection.

So $I_S^{-1}$ is well-defined and:

$\forall x \in U: I_S^{-1} \left({x}\right) = x$

Thus $I_S^{-1} \left({U}\right) = U \in \tau$.

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

$\blacksquare$


Also see


Sources