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 (n≠4) if and only if K is cellular. The case n = 4 is still
unsolved.
