Extension Theory: The interface between set-theoretic and algebraic topology

Abstract

Extension Theory can be defined as studying extensions of maps from topological spaces to metric simplicial complexes or CW complexes. One has a natural notion of an absolute (neighborhood) extensor K of X. It is shown that several concepts of set-theoretic topology can be naturally introduced using ideas of Extension Theory. Also, it is shown that several results of set-theoretic topology have a natural interpretation and simple proofs in Extension Theory. Here are sample results. Theorem. Suppose X is a topological space. Then: (a) X is normal iff every finite partition of unity on a closed subset of X extends to a finite partition of unity on X; (b) X is normal iff every countable partition of unity on a closed subset of X extends to a countable partition of unity on X; (c) X is collectionwise normal iff every partition of unity on a closed subset of X extends to a partition of unity on X; (d) if X is paracompact, then every locally finite partition of unity on a closed subset of X extends to a locally finite partition of unity on X; (e) if X is metrizable, then every point-finite partition of unity on a closed subset of X extends to a point-finite partition of unity on X. Theorem. Suppose X is a topological space. Then: (a) finite simplicial complexes are absolute neighborhood extensors of X iff every finite partition of unity on a closed subset of X extends to a partition of unity on X; (b) complete simplicial complexes are absolute neighborhood extensors of X iff every partition of unity on a closed subset of X extends to a partition of unity on X; (c) simplicial complexes are absolute neighborhood extensors of X iff every point-finite partition of unity on a closed subset of X extends to a point-finite partition of unity on X; (d) CW complexes are absolute neighborhood extensors of a first countable X iff every locally finite partition of unity on a closed subset of X extends to a locally finite partition of unity on X. Theorem. A complete simplicial complex K is an absolute neighborhood extensor of X iff its 0-skeleton K 0 is an absolute neighborhood extensor of X. Theorem. Suppose X is a topological space and A is a subset of X. Then: (a) A is C*-embedded in X iff every finite partition of unity on A extends to a finite partition of unity on X; (b) A is C-embedded in X iff every countable partition of unity on A extends to a countable partition of unity on X; (c) A is P-embedded in X iff every partition of unity on A extends to a partition of unity on X; (d) A is M-embedded in X iff every partition of unity α on A extends to a partition of unity β on X so that β(B) = α(A) for some zero-set B of X which contains A.