This is from a post in sci.math that did not get a full answer; I may repost it for the OP there:
I am interested on the issue I read in another site of when an embedding from a closed set extends into a homeomorphism, i.e., if $C$ is closed in $X$, and $f:C \rightarrow Y$ is an embedding, when can we extend $f$ to $F$ so that $F:X \rightarrow Y$ is a homeomorphism, and $F|_C=f$ (i.e., $f=F$ in $C$ )? Of course there are trivial cases like when $f$ is the identity on $C$:
I know of, e.g., Tietze Extension, and I think there are results about extending maps from a space into its compactification; I think the map must be regular (inverse image of a compact set is compact). But I don't know of any general result.
I will learn Latex as soon as I can; my apologies for using ASCII