Date: Fri, 5 Feb 1999 15:16:24 -0500 (EST)
From: Michael Cole <mmcole@math.lsa.umich.edu>
Subject: associativity

Dear Colleagues,
     I have a followup question about smash products of based spaces.  A
few weeks back I posted a question asking for the simplest example of
based spaces X,Y, and Z such that (X smash Y) smash Z and X smash (Y smash
Z) have different topologies where we use the cartesian product to make
the smash products.  Rainer Vogt cited a counterexample due to Puppe.  The
claim was that (Q smash Q) smash N abd Q smash (Q smash N) furnish the
counterexample where Q is rational numbers and N is natural numbers.
   I do not see how this counterexample could be correct if it is true
that the smash product of based k spaces is associative.  There is a well
known and easily proved fact that spaces that satisfy the first axiom of
countability are k spaces.  Clearly Q and N are first countable as are
Q times Q and Q times N.  Generally, an arbitrary quotient space of a
first countable space need not be first countable.  However, surely
Q smash Q is first countable since the neighborhoods of the basepoint
correspond to the neighborhoods of (Q times *) union (* times Q) in
Q times Q.  There must be a countable base for these neighborhoods
since (Q \times *) union (* times Q) is a countable set and
Q times Q is first countable.  Similarly, Q times N is first countable
as are (Q smash Q) times N and Q times (Q smash N).
    It seems to me that everything in sight is first countable and
therefore is a k space.  Hence the constructions for the two associated
smash products in the category of all spaces should be the same as the
constructions in the category of k spaces and therefore associativity
should hold.  Is Puppe's counterexample incorrect, or am I missing some
simple point here?

Mike Cole