I have seen here on stack exchange (in the comments) a proof along the lines of:
$X\backslash x$ is disconnected for all $x\in X$ but $Y\backslash y$ is connected for some $y\in Y$. Therefore $X$ and $Y$ are not homeomorphic.
Explicitly, how does this show the nonexistence of a homeomorphism between $X$ and $Y$?
A homeomorphism is a bijective continuous function with continuous inverse.
A continuous function's inverse maps open sets to open sets.
A connected space is the union of disjoint open sets.
Prior attempts: Write out the definitions, then ... ?