Vol. 23, No. 2, 1967

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A generalization of the Borsuk-Whitehead-Hanner theorem

D. M. Hyman

Vol. 23 (1967), No. 2, 263–271

Let A and B be metric spaces and let f : A B be a map. Suppose that X and Y are ANR’s containing A and B, respectively, as closed subsets, and consider f to be a map from A into Y . One of the results of this paper is that the question as to whether or not the adjunction space X fY is an absolute neighborhood extensor for metric pairs (or ANR if X fY is metrizable) depends only on f and not on X and Y ; that is, if X fY is an ANE (metric) and if X and Y are replaced by ANR’s Xf and Y , respectively, then x∕ fY is an ANE (metric). This result is a consequence of the main theorem: Let B be a strong neighborhood deformation retract of a space Y and suppose that both B and Y B are ANE (metric). If Y B has a certain type of covering, then Y is an ANE (metric). This generalizes the known result that if Y is metrizable, then Y is an ANR.

Mathematical Subject Classification
Primary: 54.60
Secondary: 55.00
Received: 12 December 1966
Published: 1 November 1967
D. M. Hyman