Extension theory of separable metrizable spaces with applications to dimension theory

Abstract

The paper deals with generalizing several theorems of the covering dimension theory to the extension theory of separable metrizable spaces. Here are some of the main results: Generalized Eilenberg-Borsuk Theorem. Let L be a countable CW complex. If X is a separable metrizable space and K ∗ L is an absolute extensor of X for some CW complex K, then for any map f : A → K, A closed in X, there is an extension f ′ : U → K of f over an open set U such that L ∈ AE(X − U). Theorem. Let K,L be countable CW complexes. If X is a separable metrizable space and K ∗ L is an absolute extensor of X, then there is a subset Y of X such that K ∈ AE(Y ) and L ∈ AE(X − Y ). Theorem. Suppose Gi, . . . , Gn are countable, non-trivial, abelian groups and k > 0. For any separable metrizable space X of finite dimension dimX > 0, there is a closed subset Y of X with dimGiY = max(dimGiX − k, 1) for i = 1, . . . , n. Theorem. Suppose W is a separable metrizable space of finite dimension and Y is a compactum of finite dimension. Then, for any k, 0 < k < dimW − dimY , there is a closed subset T of W such that dimT = dimW − k and dim(T × Y ) = dim(W × Y )− k. Theorem. Suppose X is a metrizable space of finite dimension and Y is a compactum of finite dimension. If K ∈ AE(X) and L ∈ AE(Y ) are connected CW complexes, then K ∧ L ∈ AE(X × Y ).