Pre-topologies and Dynamic Spaces

Approximation Spaces were introduced in order to analyse data on the basis of Indiscernibility Spaces, that is, spaces of the form 〈U, E〉, where U is the universe of data and E is an equivalence relation on U. Various authors suggested considering spaces of the form 〈U, R〉, where R is any relation. This paper aims at introducing a further step consisting in spaces of the form 〈U, {R}_i∈I〉, where {R}_i∈I〉 is a family of relations on U, that we call “Dynamic Spaces”, because they make it possible to account for different forms of dynamics. While Indiscernibility Spaces induce 0-dimensional topological spaces (Approximation Spaces), Dynamic Spacess induce various types of pre-topological spaces.