Let $X$ be a compact hausdroff toplogical space and $Y$ be topological space such that $f:X \rightarrow Y$ is bijective and continuos

Question

Which of the following option are correct 1) f is open map 2) f is a close map 3) f is both open and close 4) f is neither open, nor close

My approach: Since nothing has been said about $Y$, I assume $Y =X$ with trivial topology (i.e. Y and $\emptyset$). From this it follows that 4) is correct.

I was wondering if there is a more correct-proof based reasoning to this question?

Answer 1

Your example is right. Certainly for any Y, we can not conclude any option, but if you assume that Y is Hausdorff, then you can say 1), 2) and 3) are correct.