Upper semicontinuous decompositions of the n-sphere



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Let G be a decomposition of S[raised n], n > 1 , having only a countable number of nondegenerate elements and F a subset of S[raised n]. If F is an I-set, it is shown that it is necessary and sufficient that G = G(F) in order that G be an upper semicontinuous decomposition of S[raised n] having an n-sphere as decomposition space. From the above results it is deduced that the components of an I-set are countable. If F is a G[lowered delta]-set of S[raised 3] having only a countable number of components and G(F) is upper semicontinuous, it is shown that F is an I-set if and only if each component of F has an open 3-cell as complement. It is also shown that if F is the sum of countable number of disjoint tame arcs in S[raised 3], then F is an I-set if and only if G(F) is upper semicontinuous. Finally, using the properties of similar n-cells in S[raised n], results are obtained about upper semicontinuous decompositions of Sn and their elements which indicate when their decomposition spaces are topological n-spheres. The results of this paper parallel those of K. W. Kwun in his paper published in 1962 bearing the same title.