1 research outputs found
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.Comment: 89 page