TITLE: Do there exist two spaces such that there is no continuous, non-constant function between them?
QUESTION [5 upvotes]: 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.

REPLY [3 votes]: 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.