zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[1] Alexandroff, P., On the dimension of normal spaces, Proc. roy. soc. London ser. A, 189, 11-39, (1947) · Zbl 0038.36102
[2] Banaschewski, B., Local connectedness of extension spaces, Canad. J. math., 8, 395-398, (1956) · Zbl 0072.17703
[3] Banaschewski, B., Lectures on frames, (1988), University of Cape Town
[4] B. Banaschewski, Compactification of frames, Math. Nachr., to appear. · Zbl 0722.54018
[5] Banaschewski, B.; Mulvey, C.J., Stone-čech compactification of locales, Houston J. math., 6, 301-312, (1980) · Zbl 0473.54026
[6] Collins, P.J., On uniform connection properties, Amer. math. monthly, 78, 372-374, (1971) · Zbl 0208.50902
[7] DeGroot, J.; McDowell, R.H., Locally connected spaces and their compactifications, Illinois J. math., 11, 353-364, (1967) · Zbl 0147.41602
[8] Gilmour, C.R., Private communication, (October 1989)
[9] Henriksen, M.; Isbell, J.R., Local connectedness in the stone-čech compactification, Illinois J. math., 1, 574-582, (1957) · Zbl 0079.38604
[10] Johnstone, P.T., Stone spaces, (1982), Cambridge Univ. Press Cambridge · Zbl 0499.54001
[11] Kříž, I.; Pultr, A., Products of locally connected locales, Supp. rend. del cir. mat. di. Palermo ser. II, 11, 61-70, (1985) · Zbl 0638.54001
[12] Wallace, A.D., Extensional invariance, Trans. amer. math. soc., 70, 97-102, (1951) · Zbl 0042.16701
[13] Whyburn, T., Analytic topology, 28, (1942), Amer. Math. Soc. Colloq. Publ · Zbl 0061.39301
[14] Wilder, R.L., Topology of manifolds, 32, (1949), Amer. Math. Soc. Colloq. Publ · Zbl 0117.16204
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.