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

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.