locally compact and proper functions
i try to solve following exercise:
Let f:X-> Y be an continuous immersion. X a Hausdorff-space and Y locally compact, i.e. Y is a Hausdorff-space with the property that every point y Y has a nbh. U with closure(U) compact.
Show that f is proper iff f^(-1) (K) X is compact for all K Y compact.
"=>" This direction was not so difficult. I could solve it.
But the other direction "<=" i couldn't solve.
We have shown before, that "f is proper" is equivalent to "f is closed" and to "f is an embedding".
I try to show that f is closed, since i think this must be the most easy way.
Have you some help for me.
Thanks a lot