Locally compact semigroups with dense maximal subgroups

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motivating the study of such a problem. First, for an arbitrary locally compact semigroup, the closure of the maximal subgroup containing an idempotent forms a locally compact semigroup. If we know this maximal subgroup is a topological group, then it is a locally compact topological group. (This fact will be proved in the following paragraph.) By our rather good knowledge of locally compact topological groups, we can obtain information about the structure of this locally compact semigroup. Second, if we look at this problem from the point of view of transformation groups, we have a group G acting on a locally compact topological space. Naturally, we would like this group to be topologized so that it is a topological group. It is known that the weakest topology for G to be a topological group is the g-topology [1]. It is somewhat harder to deal with the g-topology than with the k-topology, but under some conditions the g-topology will coincide with the k-topology. Among the known conditions, the local compactness of the group seems most initeresting. If one tries to generalize such a condition, it is rather natural to ask the following: If G is a group of homeomorphisms of X onto X, and the closure of G is locally compact in the k-topology, does the induced topology of G coincide with the g-topology? Third, in the past few years there have been several papers about such questions: What is the relationship between the algebraic structure and the topological structure? When can one strengthen the topological structure so that even under weak conditions the algebraic operations will be continuous in a certain sense? For instance, in topological groups, if the group space is locally compact, and if the multiplication is separately continuous with respect to the topology of the space, then the group is a topological group. We are able to prove the same theorem when the space is second countable and second category without the assumption of local compactness [9]. Now, our present problem arises quite