Research Announcement THE SPAN OF MAPPINGS AND SPACES

Let X, Y be metric spaces, and let f: X ~ Y be a mapping. By PI and P2 we denote the standard projections of the product X x X onto X, i.e., Pl(x,x') = x and P2(x,x') = x ' for (x,x') E X x X. The span a(f) of the mapping f is the least upper bound of the set of real numbers a with the following property: there exist connected sets C C X x X a such that Pl(C) = P2(C) and a ~ dist[f(x),f(x')] for a a (x,x') E C (see [2], p. 99). The span a(X) of the space X a is the span of the identity mapping on X (see [4], p. 209). The purpose of the present paper(l) is to announce some results which relate to spans of mappings and have a nun~er of interesting consequences for spans of spaces. A complete version will be published elsewhere. The proofs of the following four propositions are rather straightforward. 1. If f: X ~ Y, then 0 < a(f) < a(Y) < diam Y. 2. If f: X ~ Y and X is aompaat, then Inf{d[f-l(y),f-l(y')]: a(f) < dist(y,y')} < a(X). 3. If f: X ~ Y, X is compact and 0 < £ < diam Y, then o < Inf{d[f-l(y),f-l(y')]: £ < dist(y,y')}. 4. If f: X ~ Y and X is compact, then a(x) o implies a(f) O. lThis paper was presented during the Thirteenth Sprin