Re: T1 topology

From: pierre.c (pierre.cussol_at_apx.fr)
Date: 10/04/04 

Date: Mon, 04 Oct 2004 13:57:53 GMT

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
>>:
>>
>>- a metric gives a separate topology according to the T2 separation
>>condition:
>>
>>For each couple of points x and y, there is an open set U which
>>contains x , an open set V which contains y and U. V is empty.
>>The triangular inequality forbids that U.V be non empty.
>>
>>- a T1 topology is not separated this way but rather :
>>
>>For each couple of points x and y, there is an open set U which
>>contains x and not y , an open set V which contains y and not x .
>>This property says nothing about U.V which can be non empty.
>>
>>So a T1 topology + a metric should be a T2 topology
>>
>>did i read properly? is it a mistake?
>
>At first it sounded like you'd read that _any_ T1 topology
>was metrizable - that's certainly wrong. But it seems like
>what you read was that some particular T1 topology was
>metrizable. There's no problem with that. Yes, it follows
>that the topology is actually T2. So what? Any T2 topology
>is also T1.
>
>>pierre.c
>
>
>************************
>
>David C. Ullrich

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?

pierre.c

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