Let X be a Tikhonov space, H(X) the group of all self-homeomorphisms of X with the usual composition and e:H(X)×X -> X, (f, x) \in f(x), the evaluation map. A group topology on H(X) which makes it a topological group is called admissible if the evaluation map is continuous. Let LH(X) be the upper semi-lattice of all admissible group topologies on H(X) ordered by the usual inclusion. The author considers the question of when LH(X) has a least element for a non-compact space X. The existence of a least element in LH(X) has been proved for T2 locally compact spaces, T2 rim-compact and locally connected spaces, and so on. Here a spaceX is called rim-compact if every point in X has arbitrarily small neighborhoods with compact boundary. In this paper the author shows that X being rim-compact is not a necessary condition in order for LH(X) to have a least element. It is known that for the set R of real numbers and the set Q of rational numbers, each with the Euclidean topology, the product R×Q is not rim-compact.Indeed, the author proves that LH(R×Q) has a least element.

### GROUP ACTION ON RxQ AND FINE GROUP TOPOLOGIES

#### Abstract

Let X be a Tikhonov space, H(X) the group of all self-homeomorphisms of X with the usual composition and e:H(X)×X -> X, (f, x) \in f(x), the evaluation map. A group topology on H(X) which makes it a topological group is called admissible if the evaluation map is continuous. Let LH(X) be the upper semi-lattice of all admissible group topologies on H(X) ordered by the usual inclusion. The author considers the question of when LH(X) has a least element for a non-compact space X. The existence of a least element in LH(X) has been proved for T2 locally compact spaces, T2 rim-compact and locally connected spaces, and so on. Here a spaceX is called rim-compact if every point in X has arbitrarily small neighborhoods with compact boundary. In this paper the author shows that X being rim-compact is not a necessary condition in order for LH(X) to have a least element. It is known that for the set R of real numbers and the set Q of rational numbers, each with the Euclidean topology, the product R×Q is not rim-compact.Indeed, the author proves that LH(R×Q) has a least element.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11386/1993122
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