Isotopy Galois spaces

Abstract

This paper introduces two classes of topological spaces, isotopy Galois spaces and orbit attraction spaces. Both classes of spaces are defined in terms of the actions of the homeomorphism group of a space onto itself, and both classes are closely related to the classes of Galois spaces and representable spaces. Although there is a metrizable Galois space X such that X x X is not a Galois space, it is shown that the product of an isotopy Galois space and any Tychonoff space is an isotopy Galois space. Consequently, isotopy Galois spaces include the total spaces of locally trivial fibre spaces with isotopy Galois base and completely regular fibre. From this result it follows that every locally compact topological group which is not totally disconnected is an isotopy Galois space. Since every locally compact totally disconnected topological group is a representable space, every non-discrete locally compact topological group is a Galois space. We make use of three types of homogeneity which were originally defined and discussed in [3] and [7] A space (X, JT) is strongly 2-homogeneous provided that for any points xl9 x2 and any points yl9 yz, there is a homeomorphism h of X onto itself such that fifa) = yt and h(x2) = y 2 A space (X, ^~) is nearly homogeneous provided that for any x e X and any D e άΓ there is a y e D and a homeomorphism h of X onto itself such that h(x) — y. A space (X, J7~) is strongly locally homogeneous if for every neighborhood of any point x, there exists a subneighborhoo d U(x) such that for any z e U(x) there is a homeomorphism h of X onto itself such that h(x) = z and h(y) = y for each y e X — U(x). It is evident that every orbit attraction space (defined in § 3) is nearly homogeneous and we show that every representable continuum is a strongly 2-homeogeneous orbit attraction space. This result suggests the conjecture that every representable connected space is locally connected. The following two questions concerning homogeneity also remain unresolved.