I've been reading through Tammo tom Dieck's text on algebraic topology and have been trying to prove a proposition (1.2.1) of his from the first chapter. Please find below an equivalent reformulation:
Let $i\colon Y \to A$ be a bijection and $g\colon Z\to Y$ a set map. If $g$ is continuous if and only if $i\circ g$ is continuous, then $i$ is a homeomorphism.
The converse was easy to show, but for this direction, I cannot seem to get my open sets to work out correctly - it's as though I need to push an open set forward but have no means of doing so. If anyone has any hints or tips, I would greatly appreciate it!
Thank you everyone! Here is my attempt at a proof (feel free to critique):
Since our assumption holds for any such $Z$ and $g$, choose $Z=Y$ and $g=\mbox{id}_Y$, which is of course continuous. Then $i\circ \mbox{id}_Y = i$ is continuous. Similarly, choosing $Z=A$ and $g=i^{-1}$ shows that $i^{-1}$ is continuous. As a continuous bijection with continuous inverse, $i$ is a homeomorphism.
I much prefer the elegance of the solution by @HennoBrandsma in the comments above, but I would have definitely not thought of using the universal property in such a way! (I hope it will become easier to invoke for me as I study it more.)
Thank you again!