Re: Countably infinite Hausdorff topology?

From: Stephen J. Herschkorn (herschko_at_rutcor.rutgers.edu)
Date: 09/15/04 

Date: Wed, 15 Sep 2004 22:18:57 GMT

David C. Ullrich wrote:

>On Tue, 14 Sep 2004 14:04:39 GMT, "shedar" <nobody@nonesuch.com>
>wrote:
>
> [...]
>
>
>
>>would not "cut it". I think one needs
>>to invoke AC (or at least the axiom of countable choice) to get the
>>sequences mentioned above.
>>
>>
>
>Seems to me that you may well be right that some sort of AC is
>required for the argument as stated. Seems like a slightly
>silly thing to point out, because people use AC without
>mentioning it this way all the time.
>
>Seems particularly silly in this case because it's trivial
>to convert the argument into one that does not use AC
>(at the expense of converting it into a proof by contradiction):
>Assume the topology is countable and fix an enumeration of
>the open sets. Now each time we need to choose an open
>set with a certain property choose the one with that
>property that comes first in the enumeration.
>

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

It's fine by me to use AC. Much of point-set topology as we know it
depends on it. And, lately, the Platonist part of me has come to
"believe in" AC. 

-- 
Stephen J. Herschkorn                        herschko@rutcor.rutgers.edu