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In this paper an outline is given of a characterization among the Peano spaces (i.e., among the compact, locally connected continua) of those spaces that are locally euclidean in dimension 3. A characterization of the 3-sphere among Peano spaces has been given by R. H. Bing [l ] and it is his result together with the notion of a "columnar partitioning" that enables one to characterize the locally euclidean spaces. Briefly, a partitioning of a space S is a finite collection of mutually exclusive connected open sets whose union is dense in S. A sequence Gif G2, • * • of partitionings of a space is called a decreasing sequence of partitionings if d+i is a refinement of Gi (i.e., an element of Gi+i is a subset of some element of Gi) and the maximum of the diameters of the elements of Gi approaches 0 with 1/i. A partitioning is regular if each of its elements is the interior of the closure of this element. A necessary and sufficient condition that a Peano space X be a topological 3-sphere is that one of its decreasing sequences of regular partitionings have the following four properties: (1) The boundary of each element of Gi is a simple closed surface. (2) If the boundaries of two elements in UG* meet, the intersection is a 2-cell. (3) The intersection of the boundaries of three elements of Gi is 1-dimensional at each of its points. (4) If g is an element of Gi-i and F(g) its boundary (g — X if i = 1), then the elements of Gi in g may be ordered gi, • • • , gn so that

LOCALLY PERIPHERALLY EUCLIDEAN SPACES ARE LOCALLY EUCLIDEAN, II