# continuous but not homeomorphism betwen Hausdorff space and compact Hausdorff space

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?

The classical example is $$f \colon [0, \, 2 \pi) \longrightarrow S^1, \quad x \mapsto (\cos x, \, \sin x).$$
The identity from $[0,1]$ endowed with the discrete topology (every set is open) to $[0,1]$ with its usual topology does the job.