Re: Adjunction Spaces


Justin Young wrote:
> "Per Vognsen" <per.vognsen@xxxxxxxxx> wrote in message
> news:1119734311.268170.14220@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> > Justin Young wrote:
> >> I am trying to prove that the restriction of q to Y is an embedding. The
> >> proof of this when A is closed is straightforward. I'm not completely
> >> sure
> >> it's true, but have been unable to find a counterexample.
> >>
> >> Clearly q is injective and continuous. The only issue is showing that it
> >> takes closed sets in Y to closed sets in q(Y) (or open sets). I have not
> >> been able to show this. The main problem I'm having is that q(Y) is a
> >> subspace of a quotient space.
> >
> > It is a general fact that quotient maps are open, i.e. they take open
> > sets to open sets.
>
> Maybe your defintion of quotient maps differs from mine.
> A map p is a quotient map if and only if (p^-1(U) is open if and only if U
> is open.)

Of course, I'm not sure what I was thinking. After further
consideration, I'm convinced that A indeed needs to be closed for the
proposition to hold in general. I mean, if A is open and f is such that
f(A) is open in Y then won't q(Y) be non-Hausdorff in general? If you
take Y to be Hausdorff, this should constitute a counterexample.

Per

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