H-4 - Homogeneous Spaces

Abstract

In topology, for an arbitrary topological spaceX, the autohomeomorphism group Aut(X) consisting of all homeomorphism of X onto itself corresponds to such a group. Every group is isomorphic to the autohomeomorphism group of some compact Hausdorff space. A topological space X is called topologically homogeneous (a homogeneous space) provided that for arbitrary points x, y ∈X there exists a homeomorphism : X →X such that ƒ (x) is equal to y. From the viewpoint of a transformation group, it is believed that homogeneous spaces are those spaces X on which Aut(X) acts transitively. There are some extensive theories on the relation between the topological space X and its Aut(X); however, this chapter restricts its attention only to basic definitions and results in general topology. For a topological group G and its closed subgroup H, the factor space G/H is called a homogeneous space. Topological groups and more generally factor spaces of topological groups are typical examples of homogeneous spaces in the sense of topological homogeneity. In a topologically homogeneous space X, a topological property that holds at one point of X also holds at all other points.