Region-Based Topology

Axioms, Algebras and Topology

“…Thus, relevant properties for spaces models are typically weaker than better known separation properties, in that they impose conditions on regular subsets of the space rather than on points or on open or closed sets in general. Although we believe that these representations are particularly natural, it is worth noting that there may be alternative topological representations where the embedding of the algebraic structure in the topology takes a different form (as with the representations of [93] and [76]). …”

“…In the sequel, we will exhibit both positive and negative representation results for contact relations in topological spaces. Apart from the earlier topological representation results of Roeper [93] and Mormann [76], which do not result in the standard topological contact, the first "standard" representation result for a class of contact algebras was discovered by Vakarelov et al [106]. It utilises the theory of proximity spaces which have been briefly described in Section 2.5.…”

Gunk, Topology and Measure

“…These approaches divide into three categories: the measure theoretic approach (see Skyrms 1993 andSikorski 1964), the topological approach (see Roeper 1997), and the metric approach (see Gerla 1990). In this paper I will not look in any detail at the metric approach.…”

“…Following Peter Roeper (Roeper 1997), let us call a map h from a pointless physical space S to a pointless field value space VS a 'bounded continuous mereological' map, if it satisfies the following constraints:…”

“…One can prove (see Roeper 1997) that there is a 1-1 correspondence between bounded continuous mereological maps (between two pointless spaces) and continuous pointy functions (between the two corresponding locally compact T 2 spaces) which map regular closed sets of points to regular closed sets of points. That is to say, if we specify the state of a pointless field in a pointless space by means of a bounded continuous mereological map h, then this is equivalent to specifying the corresponding pointy field in the corresponding pointy space by means of a function f from points in space to pointy field values, where f must satisfy the constraint that it maps regular closed sets of points in space to regular closed sets of field values.…”