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