Shape morphisms and spaces of approximative maps

Abstract

For compact metric spaces X , Y contained in a given compact AR Q , the authors consider the set A(X,Y) of all approximative maps (in the sense of K. Borsuk [same journal 62 (1968), 223–254]). On A(X,Y) they define a metric making A(X,Y) a connected separable metric space, which contains the space of continuous mappings Y X as a closed subset. Moreover, path components of A(X,Y) coincide with homotopy classes of approximative maps. The latter property is of interest in view of the fact that these classes can be interpreted as shape morphisms from X to Y . Previously defined metrics on A(X,Y) had only the property that path connected approximative maps were homotopic, but the converse did not hold [the authors, Math. Japon. 31 (1986), no. 4, 623–633].