Re: T1 topology

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 10/04/04 

Date: Mon, 04 Oct 2004 08:15:27 -0500

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

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