Countably P-Concentrative Pairs and the coincidence index

In this paper, a new coincidence index is presented for countably P-concentrative pairs. 1. INTRODUCTION Let X and Y be metric spaces. A continuous single valued map p : Y-* X is called a Vietoris map [1,2] if the following two conditions are satisfied: (i) for each x E X, the set p-l(x) is acyclic, (ii) p is a proper map, i.e., for every compact A C X we have that p-l(A) is compact. Let D(X, Y) be the set of all pairs X ~ Z ~q Y where p is a Vietorismap and q is continuous. We will denote every such diagram by (p, q). In Section 2, we discuss the coincidence index for pairs Ü ~ Z q-* X where U is an open subset of a Fréchet space X. In particular, we use the results of [3] together with the notion of r-dominated maps (based on work of Borsuk and Granas) to present a coincidence index for pairs of the above form. Let Y and Z be topotogical spaces and V an open subset of Z. Then we say Y is r-dominated by V in Z ifthere exist a pair of continuous maps r : V-~ Y, s : Y-* V with r s = 1y. Let ANR, (respectively, AR) denote the class of metrizable absolute neighborhood retracts (respectively, absolute retracts); see [1]. The following result follows immediately from the Arens-Eelis theorem (see [4, p. 284]). THEOREM 1.1. IfY E A'NR, (respectively, AR), then Y is r-dominated by an open subset of a norme</space (respectively, by a normed space).