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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Discontinuous robust mappings are approximatable


Authors: Shu Zhong Shi, Quan Zheng and Deming Zhuang
Journal: Trans. Amer. Math. Soc. 347 (1995), 4943-4957
MSC: Primary 90C48; Secondary 49J45, 54C08
MathSciNet review: 1308024
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Abstract: The concepts of robustness of sets and and functions were introduced to form the foundation of the theory of integral global optimization. A set $ A$ of a topological space $ X$ is said to be robust iff $ {\text{cl}}A = {\text{cl}}$ int $ A$. A mapping $ f:X \to Y$ is said to be robust iff for each open set $ {U_Y}$ of $ Y$, $ {f^{ - 1}}({U_Y})$ is robust. We prove that if $ X$ is a Baire space and $ Y$ satisfies the second axiom of countability, then a mapping $ f:X \to Y$ is robust iff it is approximatable in the sense that the set of points of continuity of $ f$ is dense in $ X$ and that for any other point $ x \in X$, $ (x,f(x))$ is the limit of $ \{ ({x_\alpha },f({x_\alpha }))\} $, where for all $ \alpha $, $ {x_\alpha }$ is a continuous point of $ f$. This result justifies the notion of robustness.


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DOI: https://doi.org/10.1090/S0002-9947-1995-1308024-X
Keywords: Robust sets, robust mappings, approximatable mappings, integral global optimization
Article copyright: © Copyright 1995 American Mathematical Society