Identity Mapping is Continuous

Theorem
Let $\left({S, \tau}\right)$ and $\left({S, \tau'}\right)$ be topological spaces. topological space.

The identity mapping $I_S: (S, \tau) \to (S, \tau')$ defined as:
 * $\forall x \in S: I_S \left({x}\right) = x$

is a continuous mapping if, and only if, $\tau' \subseteq \tau$.

Proof
By definition of continuous mapping, we have that $I_S$ is continuous if, and only if

$\forall U\in \tau', I_S^{-1}(U)\in \tau$

As $I_S^{-1}(U)=U$, the previous statement is equivalent to

$\forall U\in \tau', U\in \tau$

which means

$\tau'\subseteq \tau$.