  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.
