Let be X a topological space and we suppose that E and F are homeomorphic though a map f from F to E. So I ask to me if E is open/closed when F is open/closed but unfortunatley I was not able to prove or to disprove this so that I thought to put a specific question where I ask some clarification: in particular if the result is generally false I'd like to know if it can be true we additional hypotesis, e.g. Hausdorff separability, First Countability, Metric Topology, etc....

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Let be   X a topological space and we suppose that   E and   F are homeomorphic though a map   f from   F to   E. So I ask to me if   E is open/closed when   F is open/closed but unfortunatley I was not able to prove or to disprove this so that I thought to put a specific question where I ask some clarification: in particular if the result is generally false I&#039;d like to know if it can be true we additional hypotesis, e.g. Hausdorff separability, First Countability, Metric Topology, etc....

thatuglygirlyu · Accepted Answer

Consider   E  =  (  0  ,  1  ) and   F  =      R  .Obviously,   E  ,  F  ⊂      R  . It is quite easy to show that   F and   E are homeomorphic.  F is closed and open. But   E is not closed.

Let be X a topological space and we suppose that E and F are homeomorphic though a

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