A characterization of the circle and interval
Benjamin R. Halpern · Pacific Journal of Mathematics · 1970
Consider a connected T x -space X.Take the Cartesian product of X with itself n times (n ^ 2) and then remove the generalized diagonal GD n = {(x lf , x n ) e X n \ Xι = xj for some i Φ j} thus obtaining the deleted product Z = X n -GD n .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) € R 2 ] x 2 + y 1 = 1}.On the other hand, if it is only assumed (beyond X being Ύ x 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 T x and connected and Z disconnected it is possible to define an order on X which is a total order when restricted tol~α 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 X m -GD m for all m ^ 2 (m not necessarily equal to n) are determined.In particular the number of components of X m -GD m is either (m -1)! or ml /Nl Ml where 0^N,M a^-iito* for 1 1.Proof.We will use induction on the number N of elements of