Vol. 32, No. 2, 1970

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A characterization of the circle and interval

Benjamin Rigler Halpern

Vol. 32 (1970), No. 2, 373–414

Consider a connected T1-space X. Take the Cartesian product of X with itself n times (n 2) and then remove the generalized diagonal GDn = {(x1,,xn) Xn|xi = xj for some ij} thus obtaining the deleted product Z = Xn GDn. If Z should be disconnected then a great deal can be said about X. For example, if X is compact and metrizable, then X is homeomorphic to the closed interval [0,1] or to the circle C = {(x,y) R2|x2 + y2 = 1}. On the other hand, if it is only assumed (beyond X being T1 and connected and Z disconnected) that X is Hausdorff, locally connected and separable, then X must be homeomorphic to either (0,1),(0,1],[0,1] or C. In general, without any assumptions beyond X being T1 and connected and Z disconnected it is possible to define an order on X which is a total order when restricted to X a certain finite set, and such that the order topology is coarser (weaker, smaller) then the original topology on X. Furthermore, all connected subsets of X and the components of Xm GDm for all m 2( m not necessarily equal to n) are determined. In particular the number of components of Xm GDm is either (m 1)! or m!∕N!M! where 0 N,M < n, N + M m and each of these numbers is taken on for some X satisfying our hypothesis. The “generalized” cut point behavior of X is completely determined and an interesting result is that either there are no cut points or all but at most n points are cut points.

Mathematical Subject Classification
Primary: 54.55
Received: 3 June 1968
Revised: 22 July 1969
Published: 1 February 1970
Benjamin Rigler Halpern