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Let G be an u.s.c. decomposition of S3. Let H denote the set of nondegenerate elements and P be the natural projection of S3 onto S3/G. Suppose that each point in the decomposition space has arbitrarily small neighborhoods with 2-sphere boundaries which miss P(H). We prove in this paper that this condition implies that S3/G is homeomorphic to S3. This answers a question asked by Armen trout [1, p. 15]. Actually, the hypothesis concerning neighborhoods with 2-sphere boundaries is necessary only for the points of P(H).

Decomposition spaces having arbitrarily small neighborhoods with 2-sphere boundaries II