Relaxing homeomorphism to compact domain to preserve fixed point property.

Question

I am struggling to come up with proof of the following assertion.

I have two metric spaces $X$ and $Y$ as well as a continuous (but not bi-continuous) bijection from $X$ onto $Y$. Now, if $X$ is compact and such that every continuous self-map on $X$ has a fixed point, then $Y$ also has the fixed point property.

I am afraid, I don't see how to adapt the standard proof that the fixed point property is a topological property in order to arrive at this result.

Thank you very much.

Answer 1

If $X$ is compact then any continuous bijection from $X$ to a Hausdorff space is automatically a homeomorphism (since it maps compact sets to compact sets and thus is a closed map). So, your continuous bijection $X\to Y$ must actually be a homeomorphism.