Proper Function: Homeomorphism or Not?

[Lie]

Hello!

F: X --> Y injection.

It is true that if F is proper (the inverse image of any compact set is compact) then F: X --> F(X) is a homeomorphism?

Thanks... :)

 

[quasar987]

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.

 

[Lie]

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.

Yes, I had forgotten: F to be continuous and Y (X and) to be Hausdorff. :)

Compactly generated = union of open compact ?

Thanks... :)

 

[quasar987]

http://en.wikipedia.org/wiki/Compactly_generated_space

 

[Lie]

Thanks!

I showed that Y is locally compact space and therefore is compactly generated space.

Grateful.

 

[quasar987]

You're welcome. :)