Contruction: Bijective, continuous map from one topological space $X$ into itself that is not open

Asked May 10 '19 at 11:40

Active May 10 '19 at 12:00

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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.

edited May 10 '19 at 12:00

Mefitico

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asked May 10 '19 at 11:40

DacFayd

bijektiv=bijective? – YuiTo Cheng May 10 '19 at 11:42
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In that post different topologies are conisdered on the domain and the range. OP wants the same topology. @Javi – Kavi Rama Murthy May 10 '19 at 11:58
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@KaviRamaMurthy How about Are continuous self-bijections of connected spaces homeomorphisms? – YuiTo Cheng May 10 '19 at 11:59
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@PaulFrost How about the post I'm linking to? – YuiTo Cheng May 10 '19 at 12:01
@YuiToCheng Yes, duplicate! – Paul Frost May 10 '19 at 12:02
@PaulFrost Isn't your link same as teh one posted by YuiTo Cheng? – Kavi Rama Murthy May 10 '19 at 12:08
@KaviRamaMurthy Yes. I voted to close the question and the comment was generated automatically. But YuiTo Cheng found it! – Paul Frost May 10 '19 at 12:10

Contruction: Bijective, continuous map from one topological space $X$ into itself that is not open

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