Extending continuous function to one-point compactification

[Mikeezy]

Hi,

I'm faced with the following problem:

Let X be a locally compact (but not compact) Hausdorff space and Y be a compact Hausdorff space. Let f:X->Y be a continuous function. Show that there is at most one way to extend f from the one-point compactification of X to Y.

I can't figure out how to solve this except when I assume that the extension of f is surjective, which doesn't help a whole lot. Any suggestions would be greatly appreciated.

Mike

 

[pka]

Do you know Alexandroff’s theorem: The one point compactification X* of a top-space X is compact and X is a subspace.
The space X* is Hausdorff iff X is locally compact and Hausdroff.

It seems to me as if your result would follow from that.
Although, I do not see it right off.