### Graph approximations of set-valued maps under constraints

#### Abstract

In the paper we study the existence of constrained graph approximations of

set-valued maps with non-convex values. We prove, in particular, that

any open neighbourhood of the graph of a map satisfying the so-called

topological tangency assumptions contains a graph of constrained continuous

single-valued map provided that the domain is finite-dimensional.

set-valued maps with non-convex values. We prove, in particular, that

any open neighbourhood of the graph of a map satisfying the so-called

topological tangency assumptions contains a graph of constrained continuous

single-valued map provided that the domain is finite-dimensional.

#### Keywords

Fiberwise retraction; fiberwise absolute neighbourhood extensor over B; locally n-connected space; set-valued map

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