Chapter 2.18, Problem 3.1E

Let $X$ and $X^{\prime }$ denote a single set in the two topologies $\mathcal{T}$ and ${\mathcal{T}}^{\prime }$, respectively.

Let $i:X^{\prime }\to X$ be the identity function.

Show that $i$ is continuous $\iff {\mathcal{T}}^{\prime }$ is finer than $\mathcal{T}$.