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Topological and Metric Spaces
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
Recall that every (sequentially) compact set is (sequentially) closed. According to Lemma 4.9.1 every compact (or, equivalently, sequentially compact) set in a metric space is totally bounded and therefore bounded. Thus compact sets in metric spaces are both closed and bounded. The converse, true in R $ \mathbb R $ (the Heine–Borel Theorem), in general is false. The following is an example of a set in a metric space which is both closed and bounded, but not compact.
Metric Spaces
Published in John D. Ross, Kendall C. Richards, Introductory Analysis, 2020
John D. Ross, Kendall C. Richards
Let X be a metric space, and let S ⊆ X. The set S is said to be sequentially compact if and only if every sequence of elements in S has a subsequence that converges to a limit in S.
Characterizations of robust ε-quasi optimal solutions for nonsmooth optimization problems with uncertain data
Published in Optimization, 2021
Xiang-Kai Sun, Kok Lay Teo, Xian-Jun Long
Finally, in order to obtain approximate optimality and duality results, we recall the following basic assumptions and auxiliary results (see [26] for more details). Let be a sequentially compact topological space, and let be a given function. Consider the following basic assumptions: φ is upper semicontinuous for any ; is locally Lipschitz for any ;, where the derivatives are with respect to ; is weak upper semicontinuous in .
Vectorial Ekeland variational principle for cyclically antimonotone vector equilibrium problems
Published in Optimization, 2020
Chuang-liang Zhang, Nan-jing Huang
Let be a quasi metric space. A sequence is said to be left convergent to a point if and it is denoted by . A sequence is said to be left Cauchy if for any , there exists such that for all m>n>N. We say that a quasi metric space is left complete if every left Cauchy sequence is left convergent. A nonempty subset A of X is said to be left closed if any sequence with implies that . A nonempty subset A of X is said to be left sequentially compact if any sequence , there exists a subsequence of and a point such that as .
Necessary optimality conditions for a bilevel multiobjective programming problem via a Ψ-reformulation
Published in Optimization, 2018
L. Lafhim, N. Gadhi, K. Hamdaoui, F. Rahou
Let be a sequentially compact space, and Suppose that there exists a neighbourhood U of in such that for each the function is finite on U and admits a bounded convexificator on U. If in addition is upper semicontinuous then, is a convexificator of h at .