Proper => Homeomorfismo

1. Sep 18, 2011

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... :)

2. Sep 18, 2011

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.

3. Sep 18, 2011

Lie

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

Compactly generated = union of open compact ?

Thanks... :)

4. Sep 18, 2011

quasar987

5. Sep 19, 2011

Lie

Thanks!

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

Grateful.

6. Sep 19, 2011

quasar987

You're welcome. :)