1
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.