A Note on a Cellularity Theorem by Doyle
D. G. Stewart · Proceedings of the American Mathematical Society · 1967
A continuum K in Sn is cellular if Sn\K is topologically E".In [2] Doyle gave a sufficient condition for cellularity of an arc in Sn.In this paper we show that this condition is sufficient for cellularity of dendrites and can be used to get a condition for cellularity of unions of dendrites with cellular sets in S".A continuum K in Sn is said to be (P-shrinkable if there is a point q of K and for each open set U containing q there is a closed w-cell CE U such that q lies in Int C and K meets Bd C in exactly one point.If y is a point of K different from q, we will say that K is (Pshrinkable towards y.The proofs of the following lemma and theorem are identical to those given by Doyle for Lemma 1 and Theorem 1 in [2] and, therefore, will not be given here.