Date: Sat, 06 Feb 1999 00:17:04 -0500
From: Gaunce Lewis <lglewis@syr.edu>
Subject: Re: associativity

With regard to Mike Cole's msg about Q smash Q and Q smash N:

Neither of these spaces is first countable at the basepoint.  This is
rather easy to see with Q smash N using a Cantor diagonal trick.  Think of
Q x N as the rational points on the horizontal lines y =n in the x-y plane.
 Let U be a nbhd of the basepoint in Q smash N, and let V be its inverse
image in Q x N.  Clearly V contains the rational points on the x-axis and
the rational points on each line y=n in some interval about the y-axis (the
interval can, of course, vary with n).  Assume that {U_n}_n is a countable
neighborhood basis for the topology at the basepoint in Q smash N and let
V_n be the inverse image in Q x N of U_n.  Define the open set W in Q x N
to consist of the rational points on the x-axis and, for each n,  the
rational points on an interval on the line y=n about the y-axis that is
half the length of an interval on that line contained in V_n.  Clearly, W
contains no V_n and its image in Q smash N contains no U_n.  Nevertheless,
the image of W in Q smash N is a nbhd of the basepoint.  Thus, Q smash N is
not first countable.  The argument for Q smash Q is a bit messier, but
essentially the same idea.

Best wishes,
Gaunce