Discontinuity of the identity function in topology

Question

According to a theorem I was taught, the identity function $id(x)=x$ from $(\mathbb{R}, \tau_1)$ to $(\mathbb{R}, \tau_2)$ is continuous if $\tau_1 = \tau_2$. Are there any examples of topologies $\tau_1$ and $\tau_2$ on the real line such that this doesn't hold?

Answer 1

Yes there are.

If $\tau_1$ is the standard topology and $\tau_2$ is the discrete topology, i.e., $\tau_2=\mathscr P(\mathbb R)$ (that means that every subset of $\mathbb R$ is $\tau_2-$open), then the identity map $$ I : (\mathbb R,\tau_1) \to (\mathbb R,\tau_2), $$ is not continuous since, if $U\subset\mathbb R$ is not $\tau_1-$open, while however it is $\tau_2-$open, then
$I^{-1}(U)=U\not\in \tau_1$, while $U\in\tau_2$.