As I was reading Gelfand theory, I got stuck at a proposition where I am supposed to show that A continuous function between two compact Hausdorff spaces is a Homoeomorphism if some conditions are satisfied.
But in the proof, they have written that it is sufficient to prove that the map is injective. I didn't get this point, is there any result which I don't know or am I missing something.
kindly help Thanks and regards
The map is closed: For closed subset $F$ in the domain which is compact Hausdorff, then $F$ is compact, so $A(F)$ is compact in the range which is compact Hausdorff as well, so $A(F)$ is closed.
Injective closed continuous map is a homeomorphism (onto its range).