From: "Charles H. Giffen"
Subject: Re: set product?
Date: Thu, 17 Jun 1999 14:09:12 -0400
Newsgroups: sci.math,sci.logic
To: Steve Berman
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