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I read this theorem in rudin's book:

Suppose $f$ is a continous bijection from compact metric space $X$ onto a metric space $Y$ . Then $f^{-1}$ is continuous mapping of $Y$ onto $X$

I have some questions about hypothesis of the theorem.. Clearly bijection is required for the existence of inverse. But we need compact? Do compact a necessary condition?

I want to see few counterexample,that if $X$ is not a compact metric space,then $f^{-1}$ is not continuous.

Thanks

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  • $\begingroup$ I see an example in rudin mapping [0,$2\pi)$ onto unit circle.please give me some more examples other than this $\endgroup$ – Cloud JR Oct 6 '18 at 7:41
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    $\begingroup$ $X$ discrete and $Y$ non-discrete so $X= \mathbb{N}$, $0\to 0$ and $n \to \frac1n$. $\endgroup$ – Henno Brandsma Oct 6 '18 at 7:41
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    $\begingroup$ Here's many examples $\endgroup$ – Giuseppe Negro Oct 6 '18 at 7:42
  • $\begingroup$ @Henno Brandsma then $Y={\frac{1}{n}:n\in\mathbb N}$ $\endgroup$ – Cloud JR Oct 6 '18 at 7:47
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    $\begingroup$ And $\{0\}$ too is in $Y$ $\endgroup$ – Henno Brandsma Oct 6 '18 at 8:42

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