Properties of countable separation and implicit function theorem

  • Cristina M. Di Bari
  • Pasquale Vetro


We consider some properties of countable separation for familiesA andD of subsets of a setX by means of elements of a fixed familyMP(X). We give necessary and sufficient conditions, in terms of measurability of some sets constructed by means of multifunctions, in order that the familiesA andD satisfy such a property. As an application we derive an implicit function theorem for functions ofCarathéodory-type.


Topological Space Measurable Space Closed Subset Implicit Function Theorem Countable Family 


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  1. [1]
    Averna D.,Separation properties in X and 2X.Upper semicontinuous and measurable multifunctions, Rend. Circ. Mat. Palermo38 (1989), 140–151.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    Averna D.,Separation properties in X and 2X:measurable multifunctions and graphs, Mat. Cas. (to appear).Google Scholar
  3. [3]
    Cristensen J. P. R.,Topology and Borel structure, North-Holland Amsterdam, 1974.Google Scholar
  4. [4]
    Di Bari C. M.,Measurability and countable separation, Rend. Circ. Mat. Palermo39 (1990).Google Scholar
  5. [5]
    Dravecký J.Spaces with measurable diagonal, Mat. Čas.25 (1975), 3–9.MATHGoogle Scholar
  6. [6]
    Dravecký J.—Neubrunn T.Measurability of functions of two variables Mat. Čas.23 (1973), 147–157.MATHGoogle Scholar
  7. [7]
    Hou S. H.,Implicit function theorem in topological space, Appl. Analysis13 (1982), 209–217.MATHCrossRefGoogle Scholar
  8. [8]
    Sainte-Beuve M.-F.,On the extension of von Neumann-Aumann’s theorem, J. Functional Analysis17 (1974), 112–129.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 1990

Authors and Affiliations

  • Cristina M. Di Bari
    • 1
  • Pasquale Vetro
    • 1
  1. 1.Dipartimento di Matematica ed ApplicazioniPalermo

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