A de Vries-type duality theorem for the category of locally compact spaces and continuous maps. II

Abstract

This paper is a continuation of [7], where a duality theorem for the category HLC of locally compact Hausdorff spaces and continuous maps is proved. In the present paper, we characterize the injective and surjective morphisms of the category HLC, as well as of its cofull subcategories determined by the open, by the skeletal or by the perfect maps, by means of some corresponding properties of their dual morphisms. This is in analogy with some well-known results of M. H. Stone [17] and generalizes some similar results of de Vries [4]. Such characterizations are also obtained for the homeomorphic embeddings, dense embeddings, LCA-embeddings etc.; in particular a theorem of Fedorchuk [10, Theorem 6] is generalized. Again in analogy with some well-known results of M. H. Stone [17], the dual objects of the open, regular open, clopen, compact open, regular closed etc.subsets of a locally compact Hausdorff space X are directly described by means of the dual object of X; some of these results are new even in the compact case. An explicit description of the products of local contact algebras (= LC-algebras) in the category DHLC dual to the category HLC is given and a completion theory for LC-algebras is developed.

A generalization of the Stone Duality Theorem

“…This paper is a continuation of the papers [13,15,16] and, to some extent, of the papers [7,8,9,10,11,12,14,29]. In it we prove a new duality theorem for the category of precontact algebras which implies the Stone Duality Theorem, its connected version obtained in [16], the recent duality theorems from [3,23], and some new duality theorems for the category of contact algebras and for the category of complete contact algebras.…”

On dimension andweight of a local contact algebra

“…Theorem 2.11. ( [9]) Let X be a locally compact Hausdorff space and F ∈ RC(X). Let B df = RC(X) F be the relative algebra of RC(X) with respect to F ,…”

“…Here we prove Theorem 4.4, mentioned above. We show as well that the weight of a local contact algebra is equal to the weight of its LCA-completion (see [9,7] for this notion), find an algebraic analogue of Alexandroff-Urysohn theorem for bases ([18, Theorem 1.1.15]), describe the LCAs whose dual spaces are metrizable, and characterize the LCAs whose dual spaces are zero-dimensional. Furthermore, for a dense Boolean subalgebra A 0 of a Boolean algebra A, we construct an NCA A, ρ such that w a ( A, ρ ) = |A 0 |, and if A is complete, then its dual space is homeomorphic to the Stone dual of A 0 .…”

“…Subsequently, Dimov [8] extended de Vries' duality from the category HC to the category HLC of locally compact Hausdorff spaces and continuous maps, and, on the base of this result, he also obtained an extension of Stone's duality from the category Stone of compact zero-dimensional Hausdorff spaces and continuous maps to the category ZHLC of zero-dimensional locally compact Hausdorff spaces and continuous maps (see [10,7]).…”

“…Here we prove Theorem 3.4 and Corollary 3.5, mentioned above. It is shown as well that the dimension of a normal contact algebra is equal to the dimension of its NCA-completion (see [9,7] for this notion), that the dimension of any NCA of the form B, ρ s (where ρ s is the smallest contact relation on B) is equal to zero (as it should be), and that the dimension of every relative LCA of an LCA B, ρ, B is smaller or equal to dim a ( B, ρ, B ). Recall that L. Heindorf (cited in [23]) introduced the notion of A-dimension for Boolean algebras, where A is an arbitrary non-empty class of Boolean algebras.…”