Hyper-continuous convergence in function spaces. (English) Zbl 1066.54019

Di Maio, Giuseppe; Meccariello, Enrico; Naimpally, Somashekhar

Quest. Answers Gen. Topology 22, No. 2, 157-162 (2004).

Summary: Let \((X, d)\), \((Y, e)\) be metric spaces, \(f_n\), \(f\in C(X, Y)\). As is well known, \(f_n\) converges continuously to \(f\) iff for each sequence \(x_n\to x\) in \(X\), \(f_n(x_n)\to f(x)\) in \(Y\). A beautiful result of Kuratowski states that if \(X\) is compact, the sequence \(f_n\) converges uniformly to \(f\) if and only if \(f_n\) converges continuously to \(f\). In this paper, we generalize the above result to convergences in hyperspaces. As an example, let \(d_H\), \(e_H\) be Hausdorff pseudo-metrics on the power sets of \(X\), \(Y\), respectively. The convergence \(f_n\to f\) is called Hausdorff-continuous-convergence iff for any nonempty subsets \(A_n\), \(A\) of \(X\), \(d_H(A_n, A)\to 0\) implies \(e_H(f_n(A_n),f(A))\to 0\). The following are equivalent:
(a) \(X\) is a UC space i.e. every continuous function on \(X\) to a uniform space is uniformly continuous;
(b) uniform convergence \(\Rightarrow\) Hausdorff-continuous-convergence in \(C(X,Y)\);
(c) uniform convergence \(\Leftrightarrow\) Hausdorff-continuous-convergence in \(C(X,Y)\).

MSC:

54C35	Function spaces in general topology
54B20	Hyperspaces in general topology
54E05	Proximity structures and generalizations
54E15	Uniform structures and generalizations
54A20	Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)

Keywords:

function space topologies; continuous convergence; uniform convergence; proximal convergence; set-open topology; compact convergence; metric spaces; UC space; Hausdorff metric