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- W2907996986 abstract "Introduction. The objective of this paper is to present a normed linear lattice characterization of C(X) for completely regular spaces X which have locally connected Stone-Cech compactifications. Characterizations of C(X) for completely regular spaces X which have disconnected, basically disconnected, and extremally disconnected Stone-Cech compactifications are known, and can be found in [4]. The characterization we present provides elementary proofs for several known results concerning locally connected compactifications. These include the theorem due to Banaschewski (for the sufficiency) and Henriksen and Isbell (for the necessity) that a completely regular space has a locally connected Stone-Cech compactification if and only if it is locally connected and pseudocompact. All topological spaces considered in this paper are assumed to be completely regular. Normed linear lattices are often called normed vector lattices in the literature. No distinction is intended here. An announcement of this paper containing the characterization lemma was presented at the International Symposium on Extension Theory of Topological Structures, held in Berlin in August 1967, and will appear in the proceedings of this Symposium. The key lemma for the proof of Theorem 3 was established by Professor A. J. Ward and is presented here (Lemma 5) essentially without change. Professor Ward also supplied a fundamental counterexample presented here (Example 9). It is a pleasure to acknowledge Professor Ward's assistance." @default.
- W2907996986 created "2019-01-11" @default.
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- W2907996986 date "1969-02-01" @default.
- W2907996986 modified "2023-09-24" @default.
- W2907996986 title "A characterization of $C(X)$ for locally connected $X$" @default.
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- W2907996986 doi "https://doi.org/10.1090/s0002-9939-1969-0240604-5" @default.
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