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There is a theorem that says that if $X$ is compact and $Y$ is Hausdorff, then the continuous map $f: X \rightarrow Y$ is a homeomorphism. I'm trying to come up with an example of a continuous one-to-one mapping where $Y$ is Hausdorff, but $X$ is not compact. I came up with an example where $X$ is compact: let's take $X$ = $[0,1]$ with a discrete topology, and $Y = [0,1]$ with an anti-discrete topology. Then the mapping will be bijective, continuous, but not a homeomorphism. Can anyone help with the first case?

VGT
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    $f(x) = 2\cdot \arctan(x)$, the function $f:\Bbb R \to (-\pi,\pi)$ is one to one because is bijective and the codomain is Hausdorff. Do you need something like this or I misinterpreted your question? – Turquoise Tilt Dec 14 '23 at 21:13

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