Q:

Question about bijective continuous mappings

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!

A:

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).

