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Usco selections of densely defined set-valued mappings

Published online by Cambridge University Press:  17 April 2009

Warren B. Moors  and
Sivajah Somasundaram
Warren B. Moors
Affiliation:
Department of Mathematics, The University of Waikato, Private Bag 3105, Hamilton, New Zealand e-mail: moors@math.waikato.ac.nz, ss15@math.waikato.ac.nz
Sivajah Somasundaram
Affiliation:
Department of Mathematics, The University of Waikato, Private Bag 3105, Hamilton, New Zealand e-mail: moors@math.waikato.ac.nz, ss15@math.waikato.ac.nz
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Abstract

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A set-valued mapping Φ : X → 2Y acting between topological spaces X and Y is said to be “lower demicontinuous” if the interior of the closure of the set Φ−1(V): = {x ∈ X : Φ(x) ∩ V ≠ ∅} is dense in the closure of Φ−1(V) for each open set V in Y. Čoban, Kenderov and Revalski (1994) showed that for every densely defined lower demicontinuous mapping Φ acting from a Baire space X into subsets of a monotonely Čech-complete space Y, there exist a dense and Gδ subset X1 ⊆ X and an usco mapping G: X1 → 2Y such that G (x) ⊆ Φ*(x), for every x ∈ X1, where the mapping Φ*: X → 2Y is the extension of Φ defined by, W is a neighbourhood of x}.

In this paper we present a proof of the above result with the notion of monotone Čcech-completeness replaced by the weaker notion of partition completeness. In addition, we observe that if the range space also lies is Stegall's class then we may assume that the mapping G is single-valued on X1.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 65 , Issue 2 , April 2002 , pp. 307 - 313
Copyright
Copyright © Australian Mathematical Society 2002

References

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