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_ !!! ...