Approximate extension property of mappings
The problem of extension of mappings from closed subsets of a given space into nice spaces such as a CW-complex, a polyhedron or an ANR-space over the whole space, is one of the interesting problems of geometric topology. It was studied by many mathematicians (A. Dranishnikov, J. Dydak, I. Ivanšic and L. Rubin and others).NEWLINENEWLINEIn the present paper the author investigates this problem for mappings whose codomains are more general spaces. For this aim he gives a generalization of the extension property and defines the notions of approximate extension property, absolute approximate extensor, approximate extension type, and approximate extension dimension and discusses their well-definedness.NEWLINENEWLINEUsing auxiliary lemmas and propositions obtained in the paper, the author proves that shape equivalent spaces have the same extension type and for each metrizable space \(X\), there exists a minimal element in the class of all approximate extension types of finistic separable metrizable spaces \(Y\) which are absolute approximate extensions of \(X\). Besides, the wedge theorem for absolute approximate extensors is proved, properties of closed mappings whose fibers are absolute approximate extensors are studied and the relation of the approximate extension property to shape retracts is established.
scientific article; zbMATH DE number 6023708
scientific article