Re: New countable infiniity logic

From: Shmuel (Seymour J.) Metz (spamtrap_at_library.lspace.org.invalid)
Date: 11/22/04 

Date: Mon, 22 Nov 2004 18:59:35 -0500

In <REM-2004nov21-008@Yahoo.Com>, on 11/21/2004
   at 01:18 PM, rem642b@Yahoo.Com (tinyurl.com/uh3t) said:

>I can't find the definition online, and am not sure I remember
>"separable", but wouldn't that be something like:

>For rationals and reals in a totally ordered set, reals are separable
>by rationals iff for every two different reals R < R', there is a
>rational r, such that R < r < R'.

No. Any dense subset of the reals would have that property. Separable
means that there is a countable dense subset.

>For rationals and reals in an abstract topological space,

The elements of an abstract topological space are not, in general,
reals.

>reals are separable iff for every two different reals R and R',
>there is an open set which contains R but not R'.

No, that's T1. The space is separable if it has a countable dense
subspace. 

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Shmuel (Seymour J.) Metz, SysProg and JOAT  <http://patriot.net/~shmuel>
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