Re: T1 topology

From: shedar (nobody_at_nonesuch.com)
Date: 10/05/04 

Date: Tue, 05 Oct 2004 04:45:14 GMT

"David C. Ullrich" <ullrich@math.okstate.edu> wrote in message
news:4se3m0h5he9v7u1m25ei9sl2on1n1ajt7r@4ax.com...
> On Mon, 04 Oct 2004 13:57:53 GMT, pierre.cussol@apx.fr (pierre.c)
> wrote:
>
> >On Mon, 04 Oct 2004 08:15:27 -0500, David C. Ullrich
> ><ullrich@math.okstate.edu> wrote:
> >
> >>On Mon, 04 Oct 2004 12:37:55 GMT, pierre.cussol@apx.fr (pierre.c)
> >>wrote:
> >>
> >>>I read that a T1 topology is metrizable . I do not understand because
> >>>:
> >>>[...]
> >
> >I found this
> >
> >http://mathworld.wolfram.com/T1-Space.html
> >
> >Wher it is said that T1 spaces are complete and metrizable but with
> >the restriction that the space be locally convex.
> >
> >Do you know where i can find a clue?
>
> Right here. One clue: Don't believe what you read at mathworld.
> It's full of errors - that page is one of the worst I've seen.
> As Edgar said, the authors are confusing two totally different
> notions that just happen to be described by the same word.
>
> >pierre.c
>
>
> ************************
>
> David C. Ullrich

Yes, that is a terribly worded page! It uses the word "Frechet" in two
different contexts without being explicit about it (assuming the author is
even aware of it). In a "pure" topological setting, "Frechet" is often used
as a synonym for "T1", but in functional analysis, a "Frechet" space is an
example of a "special" kind of complete topological vector space (such as
one whose topology is induced by some countable family of semi-norms).

Indeed, being "T2" alone is not sufficient to guarantee metrizability, let
alone being just "T1". A well-known example of a T2 (in fact, completely
normal) space which is not pseudometrizable is the Sorgenfrey line <R,T>
(aka, Right Half-Open Interval Topology), where R is the set of all real
numbers, and T is the topology generated by the collection B = { [a,b) | a,
b in R and a < b} of basic open sets.

Shedar

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