Question about bijective continuous mappings

Question

Let $X$ be a compact hausdorff topological space and let $Y$ be a topological space. Let $f: X \rightarrow Y$ be a bijective continuous mapping . Which of the statements is true

• $f$ is a closed map but not necessarily an open map
• $f$ is an open map but not a necessarily a closed map
• $f$ is both an open map and a closed map
• $f$ need not be an open map or a closed map

My attempt: Since every compact subspace of a haudorff space is closed, my answer is the first option.

Is it correct? Any hints/solution will be appreciated in case I'm wrong.

Thanks!

Answer 1

Suppose that $X=Y=[0,1]$. On $X$, consider the usual topology; on $Y$, the trivial topology. Finally, let $f(x)=x$. Then $f$ is continuous and bijective. However, it is neither an open nor a closed map. Therefore, the correct answer is d).