Continuous image of a closed and bounded set in a metric space is closed?

Question

I know that in metric space, continuous image of closed sets and continiuous image of bounded sets may not be closed or bounded. I am asking that if continuous image of a both closed and bounded set but not necessarily compact set is closed? Moreover, is it bounded? Thanks.

Answer 1

Take the metric space $(\mathbb{R}^{+}, d_{\text{discr}})$, that is, the positive real line with the discrete metric. The function

$$ \begin{align*} f: (\mathbb{R}^+, d_{\text{discr}}) &\to (\mathbb{R}, |\cdot|) \\ x &\mapsto \frac1x \end{align*} $$

is continous, as a function whose domain is a discrete space. We have that $f((0,1)) = (1,\infty).$

The interval $(0,1) \subseteq (\mathbb{R}^+, d_{\text{discr}})$ is closed and bounded, while the interval $(1,\infty) \subseteq (\mathbb{R}, |\cdot|)$ is not closed and not bounded.