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Compactification and local connectedness of frames. (English) Zbl 0722.54031
Summary: A classical result in the theory of Tychonoff spaces is that, for any such space X, its Stone-Čech compactification \(\beta\) X is locally connected iff X is locally connected and pseudocompact. Since all concepts involved in this generalize from spaces to frames, it is natural to ask whether this result already holds for the latter, and the main purpose of this paper is to show this is indeed the case (Proposition 2.3). Further, for normal regular frames, we obtain the frame counterpart of an analogous result of Wallace in terms of a certain property of covers (Proposition 3.5). Finally, we establish a number of additional results concerning connectedness which seem to be of independent interest.

MSC:
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54D05 Connected and locally connected spaces (general aspects)
06B23 Complete lattices, completions
06B30 Topological lattices
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