Homogeneous Countable Connected Hausdorff Spaces

In 1925, P. Urysohn gave an example of a countable connected Hausdorff space [4]. Other examples have been contributed by R. Bing [l], M. Brown [2], and E. Hewitt [3]. Relatively few of the properties of such spaces have been examined. In this paper the question of homogeneity is studied. Theorem I shows that there exists a bihomogeneous countable connected Hausdorff space. Theorems II and III deal with other questions related to homogeneity. A space Z is homogeneous if and only if for every pair of elements x and y of Z there exists a homeomorphism/ of Z onto itself such that f(x) =y. A space Z is bihomogeneous if and only if for every pair of elements x and y of Z there exists a homeomorphism / of Z onto itself such that f(x) = y and f(y) = x.