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.
