Pointless topology

In mathematics, pointless topology, also called point-free topology (or pointfree topology) and locale theory, is an approach to topology that avoids mentioning points, and in which the lattices of open sets are the primitive notions. In this approach it becomes possible to construct topologically interesting spaces from purely algebraic data. The first approaches to topology were geometrical, where one started from Euclidean space and patched things together. But Marshall Stone's work on Stone duality in the 1930s showed that topology can be viewed from an algebraic point of view (lattice-theoretic). Apart from Stone, Henry Wallman was the first person to exploit this idea. Others continued this path till Charles Ehresmann and his student Jean Bénabou (and simultaneously others), made the next fundamental step in the late fifties. Their insights arose from the study of "topological" and "differentiable" . Ehresmann's approach involved using a category whose objects were complete lattices which satisfied a distributive law and whose morphisms were maps which preserved finite meets and arbitrary joins. He called such lattices "local lattices"; today they are called "frames" to avoid ambiguity with other notions in lattice theory. The theory of frames and locales in the contemporary sense was developed through the following decades (John Isbell, Peter Johnstone, Harold Simmons, Bernhard Banaschewski, Aleš Pultr, Till Plewe, Japie Vermeulen, Steve Vickers) into a lively branch of topology, with application in various fields, in particular also in theoretical computer science. For more on the history of locale theory see Johnstone's overview. Traditionally, a topological space consists of a set of points together with a topology, a system of subsets called open sets that with the operations of union (as join) and intersection (as meet) forms a lattice with certain properties. Specifically, the union of any family of open sets is again an open set, and the intersection of finitely many open set is again open.