Attouch-Wets topology on Function spaces.

*(English)*Zbl 0842.54028If \((X,d)\) is a metric space then the Hausdorff metric on the space of closed, bounded subsets of \(X\) must be modified to accommodate unbounded sets. One possibility is the Attouch-Wets topology [see, for example, G. Beer and A. Di Concilio, Proc. Am. Math. Soc. 112, 235-244 (1991; Zbl 0677.54007)] which is metrizable and which can be constructed via the metric:
\[
m_d (A,B) : = \sum^\infty_{n = 1} 2^{-n} d_n (A,B)/ \bigl( 1 + d_n (A,B) \bigr)
\]
where \(d_n (A,B) : = \sup \{(x,A) - d(x,B) |: x \in B (x^*, n)\}\) and \(x^*\) is an arbitrary (but fixed) point of \(X\). This topology has advantages for handling closed, convex sets in an infinite-dimensional normed space \(X\). If \((X,d)\) and \((Y,e)\) are metric spaces then a continuous function from \(X\) to \(Y\) may be identified with its graph and the space \(C(X,Y)\) of continuous functions given a uniformity via the relative Attouch-Wets topology on the collection of graphs (viewed as closed subsets of \(X \times Y\) with the product metric \(d \times e)\).

One aim of the paper is to establish conditions that imply that a collection of functions is complete in the above uniformity. The authors prove that if \({\mathcal F}\) is a pointwise equicontinuous, pointwise bounded family of functions that is closed in \((C(X,Y), m_{d \times e})\) then \({\mathcal F}\) is complete. They give an example to show that neither the equicontinuity nor the boundedness is necessary for the conclusion. Moreover, if \((Y,e)\) is complete and has a nontrivial path then \((X,d)\) is complete if and only if every family \({\mathcal F}\) as above is complete. The ideas extend naturally, as is shown in the final section, to families of set-valued mappings (multifunctions, correspondences) from \(X\) to \(Y\) whose graphs are closed.

One aim of the paper is to establish conditions that imply that a collection of functions is complete in the above uniformity. The authors prove that if \({\mathcal F}\) is a pointwise equicontinuous, pointwise bounded family of functions that is closed in \((C(X,Y), m_{d \times e})\) then \({\mathcal F}\) is complete. They give an example to show that neither the equicontinuity nor the boundedness is necessary for the conclusion. Moreover, if \((Y,e)\) is complete and has a nontrivial path then \((X,d)\) is complete if and only if every family \({\mathcal F}\) as above is complete. The ideas extend naturally, as is shown in the final section, to families of set-valued mappings (multifunctions, correspondences) from \(X\) to \(Y\) whose graphs are closed.

Reviewer: A.C.Thompson