Graph convergence of set-valued maps and its relationship to other convergences (Q2706912)

The authors introduce the concept of even-outer-semicontinuity and provide interesting results by comparing it with closely-related ideas to be found in [\textit{A. Bagh} and \textit{R. J.-B. Wets}, Set-Valued Anal. 4, No. 4, 333-360 (1996; Zbl 0884.54011), and \textit{S. Kowalczyk}, Demonstr. Math. 27, No. 1, 79-87 (1994; Zbl 0824.54011)]. A net \(\{F_\sigma :\sigma\in \Sigma\}\) of set-valued maps from a topological space \(X\) to a uniform space \(Y\) is evenly outer-semicontinuous if, for each \(x\) in \(X\), every entourage \(V\) and element \(y\) of \(Y\), there is a neighbourhood \(U_x\) of \(x\), a neighbourhood \(U_y\) of \(y\) and \(\sigma_0\) in \(\Sigma\) such that, for every \(z\) in \(U_x\) we have \(F_\sigma(z) \cap U_y\subseteq \bigvee [F_\sigma (x)]\). In particular, they prove that, for a non-discrete \(X\), a uniform space \(Y\) is compact (resp. locally compact) if and only if every net of set-valued maps from \(X\) to \(Y\) is asymptotically upper equicontinuous (resp. equi-outer semicontinuous) when and only when it is evenly-outer-semicontinuous. They also relate even-outer-semicontinuity to graph and pointwise convergence of nets of set-valued maps.