
Re: Comparing Compactifactions
Posted:
Mar 20, 2013 10:30 PM


On Wed, 20 Mar 2013, David C. Ullrich wrote: > >> > >> >Let (f,X) and (y,Y) be compactifications of S. > >> >Assume h in C(Y,X) and f = hg. > >> > > >> >Thue h is a continuous surjection and when Y is Hausdorff > >> >a closed quotient map. > >> > > >> >k = hg(S):g(S) > f(S) is a continuous bijection. > >> >It it a homeomorphism? If so, what's a proof like? > > > >I'll have to check this out, but isn't > >k = fg^1:g(S) > f(S) a homeomorphism? > > That seems clear. Don't know what I was thinking, sorry. > Actually, it's easy to check as f,g are embeddings, g:S > g(S) and f:S > f(S) are homeomorphisms so are g^1:g(S) > S and fg^1:g(S) > f(S).
In addition, as f = hg, fg^1 = hgg^1 = h.id_g(S) = hg(S).

