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

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