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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
where Bx $ \mathcal B _x $ and By $ \mathcal B _y $ denote bases of neighborhoods of x in X and y in Y, respectively, we generate on X×Y $ X \times Y $ a topology called the product topology of topologies on X and Y. Of course, the Cartesian product X×Y $ X \times Y $ , as any set, can be supplied with a different topology, but the product topology is the most natural one and we shall always assume that X×Y $ X \times Y $ is supplied with this topology, unless explicitly stated otherwise. We leave as an exercise proof of the following simple result.
On uniform regularity and strong regularity
Published in Optimization, 2018
R. Cibulka, J. Preininger, T. Roubal
Basic notation. The distance from a point x to a subset A of a metric space is . The closure and the interior of A is denoted by and , respectively. Given sets C, , the excess of C beyond D is defined by . We use the convention that and as we work with non-negative quantities we set . The closed ball centred at a point with a radius r>0 is denoted by . A set is locally closed at its point x if there is r>0 such that the set is closed. Any singleton set will be identified with its only element, that is, we write a instead of . By we denote a set-valued mapping between sets X and Y , its graph, domain, and range are the sets , , and , respectively. The inverse of F is a mapping . We write to emphasize that the mapping f is single-valued. The space of all single-valued linear continuous operators acting between Banach spaces X and Y is equipped with the standard operator norm and denoted by . The space is equipped with the Euclidean norm, while the Cartesian product of two or more spaces is considered with the box (product) topology. By a.e. we mean almost every in terms of the Lebesgue measure.