Vol. 16, No. 2, 1966

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An application of a family homotopy extension theorem to ANR  spaces

Arthur H. Kruse and Paul William Liebnitz, Jr.

Vol. 16 (1966), No. 2, 331–336

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.

Mathematical Subject Classification
Primary: 55.40
Received: 12 October 1964
Published: 1 February 1966
Arthur H. Kruse
Paul William Liebnitz, Jr.