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Topic: Comparing Compactifactions
Replies: 6   Last Post: Mar 20, 2013 10:30 PM

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William Elliot

Posts: 2,637
Registered: 1/8/12
Re: Comparing Compactifactions
Posted: Mar 20, 2013 10:30 PM
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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 = h|g(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) = h|g(S).

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