We know a continuous bijection from a compact space to a Hausdorff space is always a homeomorphism.
But I am wondering what happens if we switch the domain and codomain. Is a continuous bijection function from a Hausdorff space to a compact space a homeomorphism?
What I think this is not true. Consider the example $f:\mathbb{R}\to [-\frac{\pi}{2},\frac{\pi}{2}]$ defined by $f(x)=\tan(x)$, then $f$ is a continuous bijection but $f^{-1}$ is not continuous.
Am I right? Thank you for any comments.
Best Answer
My attempt:
Consider the identity map $i:(\mathbb R,\tau_d)\rightarrow (\mathbb R,\tau_f)$
$(\mathbb R,\tau_d)$ is the real line with discrete topology and $(\mathbb R,\tau_f)$ is real line with cofinite topology.One is compact and other is not