An extension of de Vries duality to completely regular spaces and compactifications (Q670153)

A well-known theorem of Smirnov asserts that compactifications of a completely regular space $X$ can be described internally by means of proximities on $X$ compatible with the given topology. In 1962 H. de Vries has found an alternative proof of Smirnov's theorem by providing an algebraic description of the category \(\mathsf{KHaus}\) of compact Hausdorff spaces. Then he proved that \(\mathsf{KHaus}\) is dually equivalent to the category of complete Boolean algebras with a proximity on them, known as de Vries algebras. The goal of this article is to extend ``de Vries duality'' from the setting of compact Hausdorff spaces to that of completely regular spaces. \par The authors are doing this by introducing a category \(\mathsf{Comp}\), whose objects are compactifications $e:X\to Y$, where $X$ is a completely regular space, $Y$ is compact Hausdorff and $e$ is a dense embedding. For compactifications $e:X\to Y$, $e':X'\to Y'$, a morphism in this category is given by a pair $(f,g)$ of continuous maps with $f:X\to X'$, $g:Y\to Y'$ such that $g\circ e=e'\circ f$. After recalling Tarski duality, the category \(\mathsf{DeVe}\) of de Vries extensions is introduced, and then it is pointed out that \(\mathsf{Comp}\) and \(\mathsf{DeVe}\) are dually equivalent, thus generalizing de Vries and Tarski dualities. The last section of this paper deals with so-called maximal de Vries extensions, and it is shown that they correspond to Stone-Čech compactifications. As a corollary the authors derive that the category \(\mathsf{CReg}\) of completely regular spaces is dually equivalent to the full subcategory \(\mathsf{MDeVe}\) of \(\mathsf{DeVe}\) consisting of maximal de Vries extensions.