If $X$ is paracompact then $f(X)$ not need to be paracompact

Question

If $X$ is paracompact and $f:X \to Y$ is contínuous then $f(X)$ need not to be paracompact.

I've found this exercise at Munkres. My effort is the following:

I've proved that every discrete space is paracompact (since it is metrizable). My idea is define a function $f: X \to Y$ where $X$ is discrete (so, it is continuous), but I cannot find such function the way $f(X)$ is not paracompact. The only non paracompact space that I know is $S_\Omega \times \overline{S_\Omega}$, because it is Hausdorff, but not normal.

I don't know, maybe I can considerer the identity $I_d: S_\Omega \times \overline{S_\Omega} \to S_\Omega \times \overline{S_\Omega}$, where the topology on the domain is discrete.

Can you help me?

Answer 1

Take any topological space $X$ which is not paracompact consider the identity map from $X$ endowed with the discrete topology into $X$ with its original topology. It's continuous and onto.