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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Applications of the $ u$-closure operator

Authors: M. Solveig Espelie, James E. Joseph and Myung H. Kwack
Journal: Proc. Amer. Math. Soc. 83 (1981), 167-174
MSC: Primary 54D35
MathSciNet review: 620006
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Abstract: Let $ {\text{c}}{{\text{l}}_u}(A)$ be the $ u$-closure of a subset $ A$ of a space. We prove that a space is compact if and only if for each upper-semicontinuous multifunction $ \lambda $ on the space, the multifunction $ \mu $ defined on the space by $ \mu (x) = {\text{c}}{{\text{l}}_u}(\lambda (x))$ assumes a maximal value under set inclusion. We also prove that in a Urysohn-closed space any two subsets with disjoint $ u$-closures are separated by disjoint open subsets. The quotient space induced by identifying those points with identical $ u$-closures is investigated and shown to be $ {T_0}$.

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Keywords: Compactness, multifunctions, $ u$-closure
Article copyright: © Copyright 1981 American Mathematical Society

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