Undergrad Homeomorphism onto a not open image in the target

[cianfa72]

TL;DR

An example of homeomorphism onto the image that isn't open in the target

Can you provide an example of homeomorphism onto the image φ:U→φ(U) where the image φ(U)⊂M is not open in M w.r.t. its assigned topology ?

Thanks.

 

[martinbn]

TL;DR Summary: An example of homeomorphism onto the image that isn't open in the target

Can you provide an example of homeomorphism onto the image φ:U→φ(U) where the image φ(U)⊂M is not open in M w.r.t. its assigned topology ?

Thanks.

Take a subset $U\subset M$, which is not open and the identity map.

 

[cianfa72]

Take a subset $U\subset M$, which is not open and the identity map.

Ah ok, you mean a not open $U \subset M$ topologized with the subspace topology from $M$ taken as domain of the identity map $I_d$ (homeomorphism onto the image $I_d(U)$ w.r.t. the subspace topology from $M$).