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On the Borel class of multivalued functions of two variables
by
Grazyna Kwiecinska
University of Gdansk, Institute of Mathematics, Wita Stwosza 57, 80-952 Gdansk, Poland
Various results were published about the Borel classification of multivalued functions depending on one variable. Obviously each multivalued function of two variables x and y may be treated as a multivalued function of a single variable (x, y). The essential difference is the possibility of formulation of hypotheses concerning multivalued functions in terms of its sectionwise properties. S. Kempisty in his paper [K] has shown that if a real function of two real variables is upper semicontinuous with respect to one of its variable and lower semicontinuous with respect to the other one, then it is Borel class 1. We generalise this theorem into the case of multivalued functions in possible general abstract spaces.
A multivalued function F from a topological space (X, T(X)) to topological space (Z, T(Z) is said to be lower (resp. upper) calss \alpha iff F-(G)={x: F(x) \cap G =/= \emptyset} (resp. F+(G)={x: F(x) subset G}) is the additive Borel class \alpha whenever G is open in Z. If \alpha is 0, then a multivalued function F is called lower (resp. upper) semicontinuous.
Theorem Let (X, T(X)) be a topological space, (Y, T(Y)) a metrisable one and (Z, T(Z)) a perfectly normal topological space. If F from X×Y to Z is a multivalued function with T(X)-lower semicontinuous y-sections and T(Y)-upper semicontinuous x-sections, then F is T(X)×T(Y)-lower class 1. If moreover F has compact values, then it is also T(X)×T(Y)-upper class 1.
We will show that the topological cpaces X and Y in this theorem cannot be both entirely arbitrary.
References
[K] S. Kempisty, Sur les fonctions semi-continues par raport a chacune de deux variables, Fund. Math. 14, 1927, pp 237 - 427.
Date received: July 1, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeh-57.