Essential extension type maps for general classes of maps (Q984396)

Essentiality of admissible (or permissible) multivalued maps in completely regular topological spaces is studied. This complete regularity enables to use the Urysohn function technique. For instance, if \(B=\{x\in \overline{U} : x\in H_t(x) \text{ for some } t\in [0,1]\}\), where \(H\) is an admissible compact homotopy without fixed points on the boundary \(\partial U\) of \(U\), then we can take a function \(\mu:\overline{U}\to [0,1]\) with \(\mu|_B\equiv 1\) and \(\mu|_{\partial U}\equiv 0\), and define \(R(x):=H_{\mu(x)}(x)\). Assuming that \(H_0\) is essential, we obtain a fixed point of \(R\). Obviously, it belongs to \(B\), so \(\mu(x)=1\), and \(x\in H_1(x)\). This proves a continuation result.