TITLE: When does an Embedding extend into a Homeomorphism?
QUESTION [5 upvotes]: 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

REPLY [2 votes]: gary, I think your question is too general. For instance, there is something called 'cofibration' in topology, which deals with this type of problem under a strong condition: Namely, $f:A\rightarrow Y$ is continuous and has an extension iff every $g\colon A\rightarrow Y$ satisfies this property where $g$ is homotopic to f. 
If a space $(X,A)$ has homotopy extension property with respect to $Y$ then it is easier to check if a map $f\colon A\rightarrow X$ has an extension, because , now, there are billions of maps that should simultaneously have extensions. However, even under this condition, there is no general theorem that I know.