continuous but not homeomorphism betwen Hausdorff space and compact Hausdorff space

Question

I am searching for an example of a Hausdorff space $(X,\tau)$ and a compact Hausdorff space $(Y,\sigma)$ with a continuous bijection $f:X\rightarrow Y$ which is not a homeomorphism.

First of all I wrote

$$\mathbb{R}^*_+ \rightarrow [0,1] $$ $$x \rightarrow \frac{1}{x}$$ which is homeomorphism, but i couldnt get any idea about how the answer can be, any help please?

Answer 1

The classical example is $$f \colon [0, \, 2 \pi) \longrightarrow S^1, \quad x \mapsto (\cos x, \, \sin x).$$

This is clearly a continuous bijection, but it is not a homeomorphism since it is not an open map.

Answer 2

The identity from $[0,1]$ endowed with the discrete topology (every set is open) to $[0,1]$ with its usual topology does the job.

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Of course, the example of Francesco Polizzi is much nicer and much more important. The advantage of such "trivial" examples is rather to see very quickly and easily that a certain statement cannot be true.