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_ !!! ...