The first of the writers, on
p. 206 of Introduction to the Theory of Block Assemblages and Related Topics in
Topology, NSF Research Report, University of Kansas, 1956, defined a clean-cut pair
to be any pair (X,A) in which X is a metrizable space, A is a closed subset of X, A
is a strong deformation neighborhood retract of X, and X − A is an ANR. It is
shown in the present paper that for each clean-cut pair (X,A), X is an
ANR if and only if A is an ANR. A consequence is that for each locally
step-finite clean-cut block assemblage (cf. the report cited above), the underlying
space is an ANR. One of the central tools is a family homotopy extension
theorem.
