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  has given strong necessary conditions for βX to be locally connected, and Wallace  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).
© 1957 Department of Mathematics, University of Illinois at Urbana-Champaign
Henriksen, Melvin, and J.R. Isbell. "Local connectedness in the Stone-Cech compactification." Illinois Journal of Mathematics 1:4 (1957): 574–582.