I Homeomorphism onto a not open image in the target

cianfa72
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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.
 
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cianfa72 said:
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.
 
martinbn said:
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##).
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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