Many authors have developed the topological degree theory and the fixed point theory for set-valued maps using homological techniques (see for example [19, 28, 27, 16]).
Lately, an elementary technique of single-valued approximation (on the graph) (see [11, 1, 13, 5, 9, 2, 6, 7]) has been used in constructing the fixed point index for set-valued maps with compact values (see [21, 20, 4]).
In [20, 4] authors consider set-valued upper semicontinuous maps with compact proximally ∞-connected values. Making use of their ideas we prove new continuous approximation theorems and present a topological degree theory for u.s.c. maps with closed proximally ∞-connected values in Euclidean spaces. Thus we generalize the earlier known results (see [22, 12, 23]).
One of the main fact which permits us to construct the topological degree is the bijection (Theorem 3.6, Section 3). In Section 4 we define the class of set-valued maps appropriate to a topological degree theory (that is, for which the bijection theorem holds). For this class the definition of a degree can be reduced to the single-valued case (to the Brouwer degree). Some consequences of approximation methods and some remarks are given (Section 5).
Finally, let us note that without the assumption about compactness of values we obtain a large class of set-valued maps. We show (Section 4) that, for example, it contains all u.s.c. set-valued maps φ:P∪Y with closed contractible values, where P is a finite polyhedron and Y is an ANR-space.