A LOCALLY COMPACT, HOMOGENEOUS METRIC SPACE WHICH IS NOT BIHOMOGENEOUS

Suppose S is a topological space. If, for each two points a and b of S, there is a homeomorphism of S onto S throwing a to b, then S is said to be homogeneous. If, for each two points a and b of S, there is a homeomorphism of S onto S throwing a to band b to a, then S is said to be bihomogeneous. In [2], Kuratowski gave an example of a connected metric homogeneous space which is not bihomogene ous. His example is not locally compact (nor, indeed, is it topologically complete). Here, we give a locally com pact example. The Space X. Let P denote a pseudo-arc and C(P) denote the space of all subcontinua of P. Let X denote the subspace of C(P) comprising the nondegenerate proper subcontinua of P. Theorem O. The space X is connected and ZocaZZy compact. Proof. There is a continuous collection H of proper subcontinua of P filling up P, [1]. Now, H is a continuum in X and each point of X is connected to H by an are, [1]. Thus, X is connected. Since X is an open subset of the compact space C(P), X is locally compact. 26 Cook Theopem 1. The space X is homogeneous. Ppoof. Lehner has shown [4] that, if a and b are two proper subcontinua of P, then there is a homeomorphism h of Ponto P that takes a onto b. Then h*, the homeomorphism of C(P) onto C(P) induced by h, throws X onto X and h*(a) = b. Thus, X is homogeneous. Theopem 2. The space X is not bihomogeneous. Ppoof. Let a and b be two points of X such that b is a proper subcontinuum of a. Further, let b l,b,b3,·· · be2 a sequence of proper subcontinua of a (points of X) con verging to b such that no two of them lie in the same com posant of a and no one of them lies in the composant of a containing b. There is only one arc ab in X from a to b; for each i, there is only one arc ab, from a to bi; and if tb is an arc in X from a point t ~ b of X to b such that tb n ab = b then t is a proper subcontinuum of b (i