closed connected subspace of a cartesian product

Question

Let $Y$ be a connected CW-complex and $F\subset Y\times Y$ be a closed connected subspace such that the composition $F\subset Y\times Y \rightarrow Y$ is a bijective map, where

$Y\times Y\rightarrow Y $ is given by $(y_1,y_2)\mapsto y_{1}$.

I'm looking for a (easy) example of such $Y$ and such $F$ such that the composition map $F\subset Y\times Y \rightarrow Y$ is continuous bijective but not a homeomorphism.

Answer 1

Start with any example of a continuous bijection $f:A\to B$ between connected CW spaces that is not a homeomorphism. For example, $A$ a closed half-line and $B$ a circle.

Let $Y$ be $A\times B$, choose a point $p\in B$, and let $F\subset Y\times Y$ consist of all points $((a,b),(a',b'))$ such that $b=f(a')$ and $b'=p$. This is homeomorphic to $A\times A$, so it is connected.