In message <email@example.com>, Arturo Magidin <firstname.lastname@example.org> writes >The one-point compactification is a quotient of the 2-point >compactification. Any map that extends to the one-point >compactification will necessarily extend to the 2 point >compactification.
...but not vice-versa. The only compactification to which every suitable map extends is the Stone-Cech. Some even define it by that property ( en.wikipedia.org/wiki/Stone?C(ech_compactification ). So it is hardly a "usual" property of compactifications.
Going back to the original question, there doesn't seem to be much point in empathising the embedding used unless different embeddings in the same compact space can give different compactifications. I.e. is it possible to have two compact spaces Y and Z each containing a dense subspace X such that Y and Z are homeomorphic but there is no homeomorphism between them which is the identity on X.
... and that is indeed possible. For example, consider two two-point compactifications of N. one the sum of the one-point compactifications of the odds and the evens, the other of the one-point compactifications of the primes and the composites.
IS it possible to have two homeomorphic compactifications of X such that no homeomorphism f:Y -> Z has f(X) = X ? or such that Y\X and Z\X are not homeomorphic ? -- David Hartley