Paper detail

Point-free theories of space and time

The paper is in the field of Region Based Theory of Space (RBTS), sometimes called mereotopology. RBTS is a kind of point-free theory of space based on the notion of region. Its origin goes back to some ideas of Whitehead, De Laguna and Tarski to build the theory of space without the use of the notion of point. More information on RBTS and mereotopology can be found, for instance, in \cite{Vak2007}. Contact algebras present an algebraic formulation of RBTS and in fact give axiomatizations of the Boolean algebras of regular closed sets of some classes of topological spaces with an additional relation of contact. An exhaustive study of this theory is given in \cite{DiVak2006}. Dynamic contact algebra (DCA) \cite{Vak2014} (see also \cite{Vak2010,Vak2012}) introduced by the present author, is a generalization of contact algebra studying regions changing in time and presents a formal explication of Whitehead's ideas of integrated point-free theory of space and time. DCA is an abstraction of a special \emph{dynamic model of space}, called also \emph{snapshot} or \emph{cinematographic} model and the paper \cite{Vak2014} contains the expected representation theorem with respect to such models. In the present paper we introduce a new version of DCA which is a simplified version of the definition from \cite{Vak2014} and similar to that of \cite{Vak2012}. The aim is to use this version as a representative example of a DCA and to develop for this example not only the snapshot models but also topological models and the expected topological duality theory, generalizing in a certain sense the well known Stone duality for Boolean algebras. Abstract topological models of DCAs present a new view on the nature of space and time and show what happens if we are abstracting from their metric properties.