Vol. 24, No. 3, 1968

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Pointlike subsets of a manifold

Charles O. Christenson and Richard Paul Osborne

Vol. 24 (1968), No. 3, 431–435

Morton Brown introduced the concept of a cellular subset of Sn. As a consequence of the generalized Schoenflies Theorem it is easy to show that a subset of Sn is pointlike if and only if it is cellular. In this paper the obvious generalization of the definitions of pointlike and cellular sets are made and thier relationship in a manifold is considered. It is easy to show that a cellular subset of a manifold is pointlike. While it is not true that a pointlike subset of a manifold is cellular, it is shown that a pointlike subset of a compact n-manifold lies in a contractible n-manifold with (n 1)-sphere boundary. As a consequence of this it is shown that K is a pointlike subset of a compact n-manifold (n4) if and only if K is cellular. The case n = 4 is still unsolved.

Mathematical Subject Classification
Primary: 54.78
Secondary: 57.00
Received: 17 March 1967
Published: 1 March 1968
Charles O. Christenson
Richard Paul Osborne