It is well-known that continuous and bijective function from a compact space onto a T2 space is always a homeomorphism. What I'm trying to do is to show that this isn't true if we relax, even a little, the separation property. In particular, I'm trying to construct a continuous and bijective map from a compact T1 space onto itself such that it is not a homeomorphism. Well, it seems to be much more difficult than it looked at the beginning. I'm not even able to choose a proper space.
The canonical examples of T1 spaces not T2 (of course, what we do know because of the well-known result is that the space cannot be T2) are sets equipped with cofinite or cocountable topology,and they are indeed compact. But any bijection from one of this sets onto itself is bicontinuous.
I tried using a closed interval of the line with two origins. This is a T1 non T2 space, more or less easy to work with and I could find compact subsets not closed (which is the true reason why we need a non T2 space). But a continuous bijective function from this space onto itself must send fix the origins or swap them, neither of which it's useful for what I need (I could explain it with more detail, but I think you catch the idea. Please, tell me if I'm making a mistake)
Something much similar to this happens with the uncountable Fort Space.
As for the one point compactification of the rationals, although it is,again, T1, not T2 and compact; we can prove easily that every continuous function from this space onto itself is closed, so we cannot use it either.
In the last hours, I've been trying with the product of $[0,1]$ with usual topology and $R$ with cofinite topology. In this space there are much more possibilities and I may be able to found the map I'm looking for. However, it is so difficult to build continuous functions in this space ( or maybe I'm not used enough to working with the product)
Please tell me if my reasoning until now was correct and in which direction should I proceed, if you know where could an undergraduate student find "easily" a map like this.