5
$\begingroup$

Do there exist two topological spaces $(X,\tau_X)$ and $(Y,\tau_Y)$, each with more than one point, so that no continuous non-constant function $f:X\to Y$ exists?

Edit: As Krish commented, an example of this would be to take $X$ connected and $Y$ discrete with more than one point, so now I propose the question:

Do there exist two topological spaces $(X,\tau_X)$ and $(Y,\tau_Y)$, each with more than one point, so that no continuous non-constant function $f:X\to Y$ or $f:Y\to X$ exists?

Edit 2: I added the assumption that $X$ and $Y$ have at least two points.

$\endgroup$
  • 4
    $\begingroup$ Take $X$ connected, and $Y$ discrete space with more than one points. $\endgroup$ – Krish Sep 8 '17 at 17:20
  • $\begingroup$ @Krish Of course! Thank you, I made a change on the question to make it a little harder, I think. $\endgroup$ – Sergio Enrique Yarza Acuña Sep 8 '17 at 17:36
  • 1
    $\begingroup$ Take one of $X$ or $Y$ to be empty, or a single point. $\endgroup$ – MartianInvader Sep 8 '17 at 17:51
  • 1
    $\begingroup$ @MartianInvader Any function with empty domain is continuous. $\endgroup$ – Sergio Enrique Yarza Acuña Sep 8 '17 at 18:01
  • 2
    $\begingroup$ @SergioEnriqueYarzaAcuña: Any function with empty domain is also constant. $\endgroup$ – Eric Wofsey Sep 8 '17 at 18:07
5
$\begingroup$

Here's a neat example. Take $X=[0,1]$ with the usual topology and let $Y$ be a countably infinite set with the cofinite topology. Then any continuous map $Y\to X$ is constant, since $X$ is Hausdorff but there are no disjoint nonempty open sets in $Y$. On the other hand, there are no nonconstant continuous maps $X\to Y$ either (this is tricky to prove; for instance, see this question on MO).

$\endgroup$
3
$\begingroup$

Examples similar to Eric's exist even if you insist on Hausdorff spaces. Take $X=[0,1]$ (with the standard topology) and $Y$ which is countably infinite, Hausdorff and connected (e.g. the irrational slope topology). Then every continuous map $X\to Y$ and $Y\to X$ is constant.

$\endgroup$

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.