C-12 - Extensors

Abstract

The extension problem is considered as one of the main problems of Topology. The problem involves the question as to whether a given continuous map : A→Y admits a continuous extension to a space X, where A ⊂X is a closed subset. For reasonably nice spaces X and Y, Algebraic Topology presents a solution to the extension problem in terms of a sequence of obstructions. In the General Topology, extension problems are considered in families. A notation XτY is introduced for the property that all extension problems : A→Y on X can be resolved. A class C is considered as a class of topological spaces. A space Y is called an Absolute Extensor (AE) for the class C, Y ∈ AE(C), if the property XτY holds for all X ∈ C. The condition XτY is equivalent to Y ∈ AE({X}), where {X} is the class that consists of one space X. A topological space Y is called an absolute neighborhood extensor (ANE) for the class C, Y ∈ ANE(C), if for every X ∈ C and every continuous map ƒ : A→Y of a closed subset A ⊂X there is an open neighborhood W ⊃A and a continuous extension ▪ : W →Y to W.