(De)constructing Points: From Topology to Mereology and Back

Thomas Mormann (University of the Basque Country, ), Thomas Mormann (University of the Basque Country, )

(De)constructing Points: From Topology to Mereology and Back
Abstract Points are considered as fundamental ingredients of topology spaces. For instance, a topological space is defined as a set X of “points” endowed with some “topological structure” encapsulated in the set OX of open subsets of X, OX being a subset of the power set PX of X. The set OX of open sets of a topological space has the lattice-theoretical structure of a complete Heyting algebra. As is well known, many basic concepts of topology can actually expressed without points, but using only the lattice-theoretical structure of OX only, for instance continuity and connectedness. This has led to what has been described as “pointfree topology”. Indeed, pointfree topology may be characterized as a kind of non-classical mereology based on systems of regions exhibiting the structure of complete Heyting algebras instead of Boolean algebras as is the case for classical mereology. On the other hand, given an appropriate (pointfree) Heyting mereological algebras H, it is possible to construct for H a set of ersatz points pt(H). This set pt(H) may be endowed with a canonical topological structure O(pt(H)) isomorphic to H. In this way, under some mild restrictions, topological spaces and mereological systems may be considered as equivalent.