Newsgroups: sci.math
From: hook@win25.nas.nasa.gov (Edward C. Hook)
Subject: Re: Topology? Combinatorics? Question
Date: Wed, 19 Apr 1995 12:42:59 GMT
In article <1995Apr16.151329.43823@miavx1>, jahowald@miavx1.acs.muohio.edu writes:
|> Here's an interesting question for the math fiends. Find a countable
|> topological space which is not second-countable (and thus not first-countable).
Haven't you got these reversed ? I.e., ~(1st-countable) ==> ~(2nd-countable) .
|> I (believe that I) have proven the existence of such by counting methods, and I
|> can mention a way to find them, but none that I can construct are _definable_.
A standard example of this sort is the Arens-Fort space, which is a topology
defined on the (countable) set { (m,n) | m, n nonnegative integers }. All
points except (0,0) are open; as for (0,0), a neighborhood of that point
is a set U such that, for all but finitely many choices of m, the sets
I_m = { n | (m,n) \in U } are finite. This _does_ give a topology and it's
not hard to prove that there is no countable local basis at (0,0), hence
the space is not 1st-countable.
|> (By the way, I _promise_ this isn't homework. I began the
|> "responsibility, hw, ..." thread, after all. ;) )
|> Have fun!
|> jason howald
--
Ed Hook | Coppula eam, se non posit
Computer Sciences Corporation / NAS | acceptera jocularum.
NASA/Ames Research Center | Me? Speak for my employer?...<*snort*>
Internet: hook@nas.nasa.gov | ... Get a _clue_ !!! ...