From the moment of their appearance,
techniques of set-valued mappings and hyperspaces offered a unified
language for the formulation of problems and questions within different
fields of mathematics. This language turned out to be very versatile
and contributed substantially to building a much deeper insight in many
branches of mathematics the interrelationship of which is not at all
obvious at first sight. The remarkable success of this language has
attracted the attention of many outstanding mathematicians who have
found a large variety of applications during the last decades. The present state of the art is reflected in the fact that techniques
of set-valued mappings and hyperspaces have turned out to be
instrumental not only for the formulation and interpretation of
problems but for their solution as well. Despite its impressive
theoretical achievements in the last decades, the theory of set-valued
mappings and hyperspaces still maintains links to all kinds of
applications. In fact a trend towards applications has become its
characteristic feature. Of course, the theory has taken its natural
course and has yielded many problems which, besides their independent
inner beauty, provide ties with numerous classical fields of
mathematics. Problems connected with selections
(single-valued, set-valued, continuous, measurable), factorizations
(representations via compositions) and with fixed-points of set-valued mappings,
play a central role in the theory.
A
simplified variant of the problem, given by how to construct selections,
is related to the problem of extensions of maps ― a real mathematical ever-green;
there are not many investigations that would not touch, in an explicit or implicit way,
the problem of map extension. This last problem can be presented in an abstract setting:
Let X and Y be sets, let A be a
subset of X and let g:A →
Y be a map. One looks
now for a map f:X → Y such that the
restriction f|A is equal to g. Defining a set-valued mapping Φ:X → Y
by letting Φ(x)={g(x)}
for x|A
and Φ(x)=Y for x| A, one obtains that f is an extension of g if and
only if f is a selection of Φ. This
selection is quite often subjected to additional conditions such as
properties of descriptive type (continuity, measurability, etc.) or avoiding
given sets, maybe different for the different points, i.e. for each x ∈ X, f(x)
is an element of some subset Y(x) of Y. This problem is now a typical selection problem; Φ:X → Y
is defined here by Φ(x)=Y(x). It should be mentioned that the reduction
of an extension problem to a selection one is far from being just formal,
there are many important cases where it has facilitated the final solution.
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Valentin Gutev
