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

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. m. e X is a topological space. Then:(a) ormal iff every finite partition of unity on a closed subset of X extends to a finite partition of unity on X; ormal iff every countable partition of unity on a closed subset of X extends to a countable partition of unity on X; ollectionwise normal iff every partition of unity on a closed subset of X extends to a partition of unity on X; s paracompact, then every locally finite partition of unity on a closed subset of X extends to a locally finite partition of unity on X; s metrizable, then every point-finite partition of unity on a closed subset of X extends to a point-finite partition of unity on X. m. e X is a topological space. Then:(a) 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; te 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; cial 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; plexes 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. m. lete simplicial complex K is an absolute neighborhood extensor of X iff its 0-skeleton K0 is an absolute neighborhood extensor of X. m. e X is a topological space and A is a subset of X. Then:(a) *-embedded in X iff every finite partition of unity on A extends to a finite partition of unity on X; -embedded in X iff every countable partition of unity on A extends to a countable partition of unity on X; -embedded in X iff every partition of unity on A extends to a partition of unity on X; -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.