Extension dimensional approximation theorem (Q1862098)

The main result of the paper concerns the existence of graph approximations of set-valued maps defined on the so-called \(C\)-spaces. A topological space \(X\) has property \(C\) if, for each sequence \(\{{\mathcal U}_i\}^\infty_{i=1}\) of open coverings of \(X\), there is an open convex \(\Sigma\) being a countable union of pairwise disjoint families \(\Sigma_i\) such that \(\Sigma_i\) refines \({\mathcal U}_i\), \(i= 1,2,\dots\)\ . The authors, using the concept of the extension dimension with respect to a countable CW-complex \(L\), introduce the notions of \([L]\)-connectedness and \(UV^{[L]}\)-compactum and show that given a paracompact \(C\)-space \(X\) of extension dimension less or equal \([L]\), each upper semicontinuous set-valued map with \(UV^{[L]}\)-values in a complete metric space possesses arbitrarily close graph approximations. This result generalizes by far the well-known theorems on the graph approximability of multivalued maps.