Topological Representation of Precontact Algebras and a Connected Version of the Stone Duality Theorem -- I

The notions of a {\em 2-precontact space}\/ and a {\em 2-contact space}\/ are introduced. Using them, new representation theorems for precontact and contact algebras are proved. It is shown that there are bijective correspondences between such kinds of algebras and such kinds of spaces. As applications of the obtained results, we get new connected versions of the Stone Duality Theorems for Boolean algebras and for complete Boolean algebras, as well as a Smirnov-type theorem for a kind of compact $T_0$-extensions of compact Hausdorff extremally disconnected spaces. We also introduce the notion of a {\em Stone adjacency space}\/ and using it, we prove another representation theorem for precontact algebras. We even obtain a bijective correspondence between the class of all, up to isomorphism, precontact algebras and the class of all, up to isomorphism, Stone adjacency spaces.