A nonmetrizable image of a metrizable space

Question

It is well known that a hausdorff continuous image of a compact metric space is metrizable. What is a counterexample for noncompact case?

Answer 1

Let $X$ be an infinite set, let $\mathcal{T}_\text{d}$ be the discrete topology on $X$, and $\mathcal{T}$ any non-metrizable Hausdorff topology on $X$. Then the identity mapping $\operatorname{id}_X : X \to X$ is a continuous function from $( X , \mathcal{T}_{\text{d}} )$ onto $( X , \mathcal{T} )$, and the discrete topology is metrizable (by the discrete metric).