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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
DOI: https://doi.org/10.1090/S0002-9939-1981-0620006-5
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}$.


References [Enhancements On Off] (What's this?)

  • [E-J] M.S. Espelie and J. E. Joseph, Some properties of $ \theta $-closure, Canad. J. Math. (to appear). MR 608861 (82d:54021)
  • [H$ _{1}$] L. L. Herrington, Characterizations of Urysohn-closed spaces, Proc. Amer. Math. Soc. 55 (1976), 435-439. MR 0415570 (54:3655)
  • [H$ _{2}$] -, Remarks on $ H(i)$ spaces and strongly-closed graphs, Proc. Amer. Math. Soc. 58 (1976), 277-283. MR 0415562 (54:3647)
  • [H-L] L. L. Herrington and P. E. Long, Characterizations of $ H$-closed spaces, Proc. Amer. Math. Soc. 48 (1975), 469-475. MR 0365485 (51:1737)
  • [J$ _{1}$] J. E. Joseph, On Urysohn-closed and minimal Urysohn spaces, Proc. Amer. Math. Soc. 68 (1978), 235-242. MR 0487974 (58:7553)
  • [J$ _{2}$] -, Multifunctions and cluster sets, Proc. Amer. Math. Soc. 74 (1979), 329-337. MR 524312 (80f:54011)
  • [V] N. V. Veličko, $ H$-closed topological spaces, Mat. Sb. 70 (112) (1966), 98-112; English transl., Amer. Math. Soc. Transl. (2) 78 (1968), 103-118. MR 0198418 (33:6576)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0620006-5
Keywords: Compactness, multifunctions, $ u$-closure
Article copyright: © Copyright 1981 American Mathematical Society

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