From: "Charles H. Giffen" <chg4k@virginia.edu>
Subject: Re: set product?
Date: Thu, 17 Jun 1999 14:09:12 -0400
Newsgroups: sci.math,sci.logic
To: Steve Berman <steve@ims.uni-stuttgart.de>
Keywords: Symmetric product of sets (or spaces)

Steve Berman wrote:
> 
> In a paper on natural language modeltheoretic semantics the term 'set
> product' is used for an operation that, from two sets A and B, forms
> the set of unordered pairs {a,b} such that a is in A and b is in B. I
> am unfamiliar with this usage; as far as I know, this term is
> occasionally used as an alternative name for Cartesian product, and
> also in formal language theory for the set of concatenated strings ab
> such that a is in a set of strings A and b in a set of strings B. Is
> there an established usage in set theory or elsewhere of the term 'set
> product' as described above? If not, does the operation described
> above have an established name?
> Thanks,
> --Steve Berman

If  A  and  B  happen to be the same set (or more generally if
they at least have a nonempty intersection), this product
cannot be the same as the usual (Cartesian) product.  By taking
X = A u B (union), there is little loss in considering just
the "set product" of  X  with itself (since the corresponding
critter for  A, B  will be a subset). 

In topology we call this the symmetric product,  SP^2(X), of  
X .  It is often viewed as the quotient set obtained from
X x X (the Cartesian product) by identifying  (x,y) with
(y,x).  Indeed in topology, we define the  n-th order
symmetric product analogously:  

	SP^n(X) = (X x X x ... x X)/~

where  (x_1,...,x_n) ~ (y_1,...,y_n)  provided there is an
n-th order permutation that sends the coordinates of the 
first point to the second.  When  X  is a (topological) 
space then  SP^n(X)  receives the quotient topology from
the topology of the n-fold Cartesian power of  X.  

As a space,  SP^n(X)  has some interesting properties.  For
example,  SP^n(S^2)  is homeomorphic with complex projective
n-space  CP^n,  while  SP^n(S^1)  is homotopy equivalent
to  S^1  for all  n = 1,2,...,  and  SP^n(S^1 x S^1)  is
(if I recall correctly) homotopy equivalent to 
S^1 x S^1 x CP^n for all  n = 1,2,... .

If  x_0 \in X  is some preferred point (basepoint), then there
are canonical inclusions

	SP^n(X) \subset SP^{n+1}(X)

whose union, denoted  SP^\infty(X),  is called the infinte
symmetric product of  X.  For nice enough spaces  X  (eg.
CW complexes), one has the lovely result that the  n-th
homotopy group  \pi_n(SP^\infty(X))  of  SP^\infty(X) (using 
the evident point as basepoint) is naturally isomorphic to 
the  n-th  integral homology group  H_n(X,Z)  of  X .

--Chuck Giffen