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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.