Could anyone give an example for a bijective and continuous map $X \rightarrow X$ that is not open (and thus not is a homeomorphism)? The topology on $X$ must be se same.

I suppose there must be some simple examples, but I wasn't able to find one yet.

Some things I tried and that cannot work: The topology must be infinite. Linear maps even on infinite dimensional Banach spaces cannot work (Open Mapping Theorem). Maps $\mathbb R \rightarrow \mathbb R$ in the real numbers cannot work either.

Thank you for answers.


marked as duplicate by YuiTo Cheng, Javi, Paul Frost, Community May 10 at 12:22

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  • $\begingroup$ bijektiv=bijective? $\endgroup$ – YuiTo Cheng May 10 at 11:42
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    $\begingroup$ In that post different topologies are conisdered on the domain and the range. OP wants the same topology. @Javi $\endgroup$ – Kabo Murphy May 10 at 11:58
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    $\begingroup$ @KaviRamaMurthy How about Are continuous self-bijections of connected spaces homeomorphisms? $\endgroup$ – YuiTo Cheng May 10 at 11:59
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    $\begingroup$ @PaulFrost How about the post I'm linking to? $\endgroup$ – YuiTo Cheng May 10 at 12:01
  • $\begingroup$ @YuiToCheng Yes, duplicate! $\endgroup$ – Paul Frost May 10 at 12:02

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