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A note on extremally disconnected frames. (English) Zbl 0792.54034

This paper presents the results of an investigation done on the notion of extremal disconnectedness in the context of pointless topologies – frames. The paper is divided into four sections. In Section 1, we recall some basic facts concerning frames. In Section 2, we show some basic equivalents of extremal disconnectedness. In Section 3, we develop the machinery for computing with semi-open and preopen nuclei. In Section 4, we close the paper with several characterizations of extremally disconnected frames by utilizing preopen and semi-preopen nuclei. The results obtained here are closely related to the work of D. S. Janković [Acta Math. Hung. 46, 83-92 (1985; Zbl 0597.54037)], T. Noiri [Indian J. Pure Appl. Math. 19, No. 4, 325-329 (1988; Zbl 0664.54021)] and D. Sivaraj [ibid. 17, 1373-1375 (1986; Zbl 0612.54048)].

MSC:

54G05 Extremally disconnected spaces, \(F\)-spaces, etc.
54A05 Topological spaces and generalizations (closure spaces, etc.)
06D20 Heyting algebras (lattice-theoretic aspects)
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