A selection theorem for strongly regular multivalued mappings (Q1292189)

Let \(X\) be a paracompact space, \((Z,\rho)\) a metric space and \(F:X\multimap \mathbb{Z}\) a multifunction whose values are absolute extensors and are complete. Suppose that \(F= G\circ f\), where \(f:X\to Y\) is a continuous single-valued map of \(X\) onto some finite-dimensional paracompact space \(Y\) and \(G:Y\multimap Z\) is a strongly regular multifunction, i.e. for every \(y_0\in Y\) and every covering of \(G(Y)\), \(\varepsilon\in \text{cov } G(Y)\) there exists a neighbourhood \(U(y_0) \subset Y\) such that for every \(y_1\in U(y_0)\) there exist maps \(g:G(y_0)\to G(y_0)\), \(h:G(y_1)\to G(y_1)\) and homotopies \(g_t: G(y_0)\to G(y_0)\), \(h_t: G(y_1)\to G(y_1)\) with the following properties; (i) \(\rho(g, \text{id}_{G(y_0)})< \varepsilon\), \(\rho(h, \text{id}_{G(y_1)})< \varepsilon\); (ii) \(\rho(g_t, \text{id}_{G(y_0)})< \varepsilon\), \(\rho(h_t, \text{id}_{G(y_1)})< \varepsilon\) for \(t\in [0,1]\); (iii) \(g_0= h\circ g\), \(h_0= g\circ h\), \(g_1= \text{id}_{G(y_0)}\), \(h_1= \text{id}_{G(y_1)}\). Then for every closed subset \(A\subset X\) and every selection \(r:A\to Z\) of the restriction \(F| A\) there exists an extension \(\widehat{r}:X\to Z\) such that \(\widehat{r}\) is a selection for \(F\). The local version of a theorem of this kind is also proved.