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Topologies on Closed and Closed Convex Sets
Preface. 1. Preliminaries. 2. Weak Topologies determined by Distance Functionals. 3. The Attouch--Wets and Hausdorff Metric Topologies. 4. Gap and Excess Functionals and Weak Topologies. 5. The Fell…
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Strong uniform continuity
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Distance functional and suprema of hyperspace topologies
Let CL(X) denote the nonempty closed subsets of a metrizable space X. We show that the Vietoris topology on CL(X) is the weakest topology on CL(X) such that A -→ d(x, A) is continuous for each x ε X…
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The EPI-Distance Topology: Continuity and Stability Results with Applications to Convex Optimization Problems
TLDR
It is shown, subject to some standard constraint qualifications, that the operations of addition and restriction are continuous and are applied to convex well-posed optimization problems, as well as to convergence of approximated solutions in infinite dimensional convex programming.
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A Polish topology for the closed subsets of a Polish space
Let (X, d) be a complete and separable metric space. The Wijsman topology on the nonempty closed subset CL(X) of X is the weakest topology on CL(X) such that for each x in X, the distance functional…
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METRIC SPACES ON WHICH CONTINUOUS FUNCTIONS ARE UNIFORMLY CONTINUOUS AND HAUSDORFF DISTANCE
Atsuji has internally characterized those metric spaces X for which each real-valued continuous function on X is uniformly continuous as follows: (1) the set X' of limit points of X is compact, and…
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On the convergence of subdifferentials of convex functions
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THE APPROXIMATION OF UPPER SEMICONTINUOUS MULTIFUNCTIONS BY STEP MULTIFUNCTIONS
Let P be a right rectangular parallelepiped in R and let Y be a metric space. If Γ:P->Y is an upper semicontinuous multifunction such that for each x in P the set Γ(x) is nonempty and closed, then…
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The slice topology: a viable alternative to Mosco convergence in nonreflexive spaces
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Wijsman convergence: A survey
A net 〈Aλ〉 of nonempty closed sets in a metric space 〈X, d〉 is declaredWijsman convergent to a nonempty closed setA provided for eachx εX, we haved(x, A)=limλd(x, A). Interest in this convergence…
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