Topological representation of precontact algebras and a connected version of the Stone duality theorem – II

Dimov, Georgi ; Vakarelov, Dimiter

Serdica Mathematical Journal, Tome 44 (2018) no. 1-2, pp. 031-080

Résumé

The notions of extensional (and other kinds) 3-precontact and 3-contact spaces are introduced. Using them, new representation theorems for precontact and contact algebras, satisfying some additional axioms, are proved. They incorporate and strengthen both the discrete and topological representation theorems from [11, 6, 7, 12, 22]. It is shown that there are bijective correspondences between such kinds of algebras and such kinds of spaces. In particular, such a bijective correspondence for the RCC systems of [19] is obtained, strengthening in this way the previous representation theorems from [12, 6]. As applications of the obtained results, we prove several Smirnov-type theorems for different kinds of compact semiregular T0-extensions of compact Hausdorff extremally disconnected spaces. Also, for every compact Hausdorff space X, we construct a compact semiregular T0-extension (κX, κ) of X which is characterized as the unique, up to equivalence, C-semiregular extension (cX, c) of X such that c(X) is 2- combinatorially embedded in cX; moreover, κX contains as a dense subspace the absolute EX of X.

Détail

Keywords: (pre)contact algebra, 2-(pre)contact space, 3-(pre)contact space, (C-)semiregular space, (C-)weakly regular space, (C)N-regular space, (C-)regular space, compact Hausdorff space, compact T0-extension, Stone space, (complete) Boolean algebra, Stone adjacency space, absolute, 2-combinatorial embedding, open combinatorial embedding, 54E05, 54H10, 54D80, 06E15, 03G05, 54D30, 54D10, 54D05, 54D35, 54G05, 54C25.