On the G-compactification of Products
نویسنده
چکیده
Let β G X denote the maximal equivariant compactiίication (G-com-pactίfication) of the G-space X (i.e. a topological space X, completely regular and Hausdorff, on which the topological group G acts as a continuous transformation group). If G is locally compact and locally connected, then we show that β G (XX Y) = β G X X β G Y if and only if X X Y is what we call G-pseudocompact, provided X and Y satisfy a certain non-triviality condition. This result generalizes Glicksberg's well-known result about Stone-Cech compactifications of products to the case of topological transformation groups. 1. Introduction. In this paper we prove a generalization to the case of topological transformation groups of Glicksberg's well-known result about Stone-Cech compactifications of products. Recall, that a topological space X is pseudocompact, whenever C(X) = C*(X), i.e. every continuous real-valued function on X is bounded. A convenient characterization of pseudocompactness of a completely regular Hausdorff space X is that X contains no infinite sequence of non-empty open subsets which is locally finite. Cf. [4] and, for more about pseudocompactness, [5]. Glicksberg's theorem says that if X and Y are infinite completely regular spaces, then β(X X Y) = βX X βY if and only if X X Y is pseudocompact. See [6] and also [4] and [10] for short proofs. Adopting the techniques of [4] and [10], we were able to prove (terminology will be explained in 1.1 and 2.1 below):
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