1146 MATHEMATICS: 0. G. HARROLD PROC. N. A. S.

I Berge, C., Thdorie des graphes et ses applications (Paris: Dunod, 1958). 2 Whitney, H., "Congruent graphs and the connectivity of graphs," Amer. J. Math., 54, 150-168 (1932).

LOCALLY PERIPHERALLY EUCLIDEAN SPACES ARE LOCALLY EUCLIDEAN, II* BY 0. G. HARROLD, JR.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF TENNESSEE Communicated by N. E. Steenrod, May 14, 1962 1. Introduction.-A set-theoretic problem that has been of interest for some time is that of characterizing the locally Euclidean spaces among a more general class of topological spaces. In the one- and two-dimensional cases, a possible candidate for the more general class of topological spaces has been the Peano spaces. An expository paper by Van Kampen in the first volume of the Duke Mathematical Journal summarizes some well-known results in this situation. In the higher dimensional cases, the results attained so far have had to avoid the difficulty that the Schoenflies theorem of plane topology does not extend in the most general form. One method in the case n = 3 has been to assume the existence of a family of dis- tinguished 2-spheres satisfying a system of five postulates and then prove that a sequence of partitionings exist that permit a homeomorphism between a 3-sphere and the topological space to be established.2 4 It turns out that the axiom system in reference 4 may be carried over more or less directly to the n-dimensional case. However, the proof that a compact metric continuum satisfying our axiom system is topologically the n-sphere rests on the observation that the combinatorial steps in references 5 and 4 give one a scheme for approximating in a bi-uniform way certain subdivisions of the boundary of an n + 1 simplex and the appropriately chosen subdivisions (or partitions) of the compact metric continuum X. There has been a revival of interest in problems relating to Cartesian products. Much of this may be traced to examples by Bing.3 The present paper has some connections with the existence of Cartesian products but from a different angle than that in the examples referred to. The examples (together with their refinements) show that if a Cartesian product of two spaces and one factor are each locally Euclidean, the second factor may fail to be locally Euclidean at any point. In fact, the second factor may fail to be even a homotopy manifold. The result in this form has been announced by Kwun.6 In this paper, one may think of our problem in the following light: Suppose a locally Euclidean space A is embedded in a larger space B. Will there be conditions that will guarantee that A has a neighborhood in B that is locally Euelidean? In the study of the embedding of a k-manifold in a (k + p)-manifold, the local nature of the embedding depends strongly on the concepts of local unknottedness and local peripheral unknottedness.5 The Axiom of Deformation below seems to provide a link between the problem mentioned in this paragraph and that of the Downloaded by guest on September 26, 2021 VOL. 48, 1962 MATHEMATICS: 0. G. HARROLD 1147

preceding one in that the axiom permits one to conclude quickly that each k-sphere of the family S is "locally peripherally unknotted" in the containing space X of dimension k + 1. 2. The Space X.-Let X be a compact metric space such that X contains at least two points and each point p C X has a local basis of open sets { U} whose boundaries are members of a family S of subsets of X satisfying the following re- quirements: 1. SEPARATION AXIOM. Each S C 8 is a topological n - 1 sphere that separates X irreducibly into exactly two components. 2. DEFORMATION AXIOM. If D is a topological n - 1 cell, D C S, S E 8, U a component of X/S, V an open set containing Int D, then there is an isotopy of the identity map, the identity on X/V such that F,(Int D) c U for 0 < t, where F,(x), o < t < 1 is a function realizing the isotopy. 3. UNION AxIOM. If D1, D2 are topological n - 1 cells, each a subset of some ele- ment of 8, and if D1 n D2 is an n - 2 cell common to their boundaries (or an n - 2 sphere common to their boundaries), then an element of S exists that contains D1 U D2. 4. CONTRACTION AxioM. There is a number n7, 0 < 71 < 1, such that if SC is a simplicial subdivision of S E 8 of sufficiently small mesh, U is a component of X/S and 9 = { g1, . .., gkn + I} is a columnar partitioning' of U generated by 3C, then there is a homeomorphism t of U on U, the identity on S such that if gi' = t(gi), then, letting a(x) denote the diameter of x,

S( U) 5. MAXIMALITY AXIOM. If S C S and t is a homeomorphism of X on itself, then t(S) C S. The following are easily proved consequences of the existence of a local basis at each point with the boundaries of the open sets being members of S. 1. X is connected; 2. X is locally connected. The main result is the following theorem: THEOREM. Let X be a compact metric space containing at least two points such that each point has a local basis consisting of open sets whose boundaries are members of the family S satisfying the Separation, Deformation, Union, Contraction and Maxi- mality Axioms. Then X is homeomorphic to the n-sphere. 3. Brief Outline of the Method of Proof.-A particular type of partitioning of an n - 1 sphere' is introduced that can be extended to a partitioning of an open set of which the n - 1 sphere is a boundary. This determines a product structure in a "complex" that covers a neighborhood of the n - 1 sphere. The word "complex" is put in quotes for the following reasons: Each element of the complex in the domain space is a genuine Euclidean cell and of course may be expressed as a topo- logical product of cells of lower dimensions or as a "join" product. The corre- sponding element in the range space is not known to have a Euclidean structure- that is just what is to be proved-but the intersection relations of the boundaries of the elements are indeed isomorphic to those of the corresponding elements of the domain. Downloaded by guest on September 26, 2021 1148 MA1THEMATICS: E. J. McSHANE PROC. N. A. S.

Having given a homeomorphism from the boundary of an n-simplex to an element of our distinguished family of n - 1 spheres and a columnar partitioning of a neigh- borhood of the boundary of the n-simplex, can this columnar partitioning be copied over in the range space? By the properties 1, 2, 3, and 5 of the family 8, this turns out to be possible. In order to establish the desired homeomorphism of the n- simplex onto the closure of the open set in the range space of which the element of the family is the boundary, iteration is called for. The linear structure of the n- simplex provides us with a means of doing this in our domain space and in fact the ii eration has a kind of uniformity. In the range space, the properties 1, 2, 3, and 5 provide the possibility of iteration, and Property 4 (the Contraction Axiom) pro- vides a needed means of securing uniformity since no linear structure is present to rely on. The restricted Schoenflies theorem is used extensively.' * Research supported in part by the National Science Foundation (G-8239). 'Alexander, J. W., "The Combinational Theory of Complexes," Ann. Math., 31, No. 2, 292-320 (1930). 2 Bing, R. H., "A characterization of three-space by partitionings," Trans. Am. Math. Soc., 70, 15-27 (1951). 3 Bing, R. H., "The Cartesian product of a certain non-manifold and a line is E4," Ann. Math., 70, No. 3, 399-412 (1959). 4Harrold, 0. G., "Locally peripherally Euclidean spaces are locally Euclidean," Ann. of Math., 74, No. 2 207-222 (1961). 5 Harrold, 0. G., "Local unknottedness, local peripheral unknottedness and combinatorial structures," Proc. Inst. for 3-Dimensional Topology, University of Georgia (1962), pp. 71-83. 6 Kwun, K. W., "A Consequence of a result of Andrews and Curtis," Notices Am. Math. Soc. 61T-318, 8, No. 1, 625 (1961).

WEAK TOPOLOGIES FOR STOCHASTIC PROCESSES* BY E. J. MCSHANE

UNIVERSITY OF VIRGINIA Communicated May 14, 1962 A number of mathematical procedures can be regarded as defining a transforma- tion from one set of stochastic processes to another such; for example, determining the response of a linear device to a random input or integrating a stochastic differ- ential equation. The input may be imperfectly known or intentionally over- simplified. This makes it important to know that in some sense reasonably related to experimental procedures a "small" change of input produces only a "small" change of output. We here describe a topology intended for this purpose. For each t in a set T, let x(t) = (x(t, w): CA E- Q) be a random variable, that is, a measurable real-valued function on Q, where (Q, (a, P) is a probability triple. Given this process, Q is not accessible to experimental determination; but whenever t,, . ., tk are in T, the joint distribution of x(t,), . ., x(t,) can be estimated. Let (t denote the set of all (real) stochastic processes on the set T, and let x and x* be two members of P. These may correspond to quite different probability triples. Given any ordered k-tuple (ti, . . ., tk) of points of T (k any positive integer), and Downloaded by guest on September 26, 2021