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This is a study of when and where the Stone-Čech compactification of a completely regular space may be locally connected. As to when, Banaschewski [1] has given strong necessary conditions for βX to be locally connected, and Wallace [19] has given necessary and sufficient conditions in case X is normal. We show below that Banaschewski's necessary conditions are also sufficient and may be restated as follows: βX is locally connected if and only if X is locally connected and pseudo-compact (Corollary 2.5). Moreover, the requirement that βX be locally connected is so strong that it implies that every completely regular space containing X as a dense subspace is locally connected (Corollary 2.6).


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© 1957 Department of Mathematics, University of Illinois at Urbana-Champaign

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