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Extension dimension for paracompact spaces

Authors
Journal
Topology and its Applications
0166-8641
Publisher
Elsevier
Publication Date
Volume
140
Identifiers
DOI: 10.1016/j.topol.2003.07.010
Keywords
  • Dimension
  • Extension Of Maps
  • Absolute Extensors
  • Cw Complexes
  • Paracompact Spaces
Disciplines
  • Mathematics

Abstract

Abstract We prove existence of extension dimension for paracompact spaces. Here is the main result of the paper: Theorem. Suppose X is a paracompact space. There is a CW complex K such that (a) K is an absolute extensor of X up to homotopy, (b) If a CW complex L is an absolute extensor of X up to homotopy, then L is an absolute extensor of Y up to homotopy of any paracompact space Y such that K is an absolute extensor of Y up to homotopy. The proof is based on the following simple result (see Theorem 1.2). Theorem. Let X be a paracompact space. Suppose a space Y is the union of a family { Y s } s∈ S of its subspaces with the following properties : (a) Each Y s is an absolute extensor of X, (b) For any two elements s and t of S there is u∈ S such that Y s ∪ Y t ⊂ Y u . If f :A→Y is a map from a closed subset A to Y such that A=⋃ s∈ S Int A ( f −1( Y s )), then f extends over X. That result implies a few well-known theorems of classical theory of retracts which makes it of interest in its own.

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