The discussion revolves around the conditions under which a proper injection F: X --> Y can be considered a homeomorphism, particularly focusing on the requirements of continuity and the properties of the spaces involved, such as being Hausdorff and compactly generated.
Participants generally agree on the necessity of continuity and the Hausdorff condition for Y, but there is no consensus on the implications of these conditions for establishing a homeomorphism.
The discussion does not resolve the specific definitions and implications of compactly generated spaces and their relationship to the properties of the function F.
Yes, I had forgotten: F to be continuous and Y (X and) to be Hausdorff. :)quasar987 said:You need F to be continuous and Y to be Hausdorff and compactly generated. See Corolarry 4.97 of Lee's Introduction to topological manifolds.