Abstract. When a topological group G acts on a compact space X, its enveloping semigroup E(X) is the closure of the set of g-translations, g ∈ G, in the compact space X X . Assume that X is metrizable. It has recently been shown by the first two authors that the following conditions are equivalent: (1) X is hereditarily almost equicontinuous; (2) X is hereditarily non-sensitive; (3) for any compatible metric d on X the metric d G (x, y) := sup{d(gx, gy) : g ∈ G} defines a separable topology on X; (4) the dynamical system (G, X) admits a proper representation on an Asplund Banach space. We prove that these conditions are also equivalent to the following: the enveloping semigroup E(X) is metrizable.

A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. The Roelcke uniformity (or lower uniformity) on a topological group is the greatest lower bound of the left and right uniformities. A group is Roelcke-precompact if it is precompact with respect to the Roelcke uniformity. Many naturally arising non-Abelian topological groups are Roelcke-precompact and hence have a natural compactification. We use such compactifications to prove that some groups of isometries are minimal. In particular, if U_1 is the Urysohn universal metric space of diameter 1, the group Iso(U_1) of all self-isometries of U_1 is Roelcke-precompact, topologically simple and minimal. We also show that every topological group is a subgroup of a minimal topologically simple Roelcke-precompact group of the form Iso(M), where M is an appropriate non-separable version of the Urysohn space.Comment: To appear in Topology and its Applications. 39 page

Topology 37 (1998) 791-803. doi:10.1016/S0040-9383(97)00048-7Received by publisher: 1996-10-20Harvest Date: 2016-01-04 12:20:53DOI: 10.1016/S0040-9383(97)00048-7Page Range: 791-80

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