Re: Countably infinite Hausdorff topology?

From: Chan-Ho Suh (suh_at_math.ucdavis.nospam.edu)
Date: 09/16/04 

Date: Thu, 16 Sep 2004 00:54:28 -0700

In article <ld32d.3003$6V2.2320033@news4.srv.hcvlny.cv.net>, Stephen J.
Herschkorn <herschko@rutcor.rutgers.edu> wrote:

> Aren't you using AC to make an infinite number of choices ("each time we
> need to choose...")?
>

This is a common point of confusion. You only need invoke AC to make
an infinite number of *arbitrary* choices.

The typical saying to explain this is, "You need AC to choose a sock
from each pair of an infinite sequence of pairs of socks, but you don't
need AC to choose a left shoe from each pair of an infinite sequence of
pairs of shoes."

AC, in its most common guises, asserts the existence of a choice
function. When you can explicitly define a choice function (like in
the shoe example), you don't need to resort to AC to show the existence
of one.

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