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Pointless topology

Topological approach that uses lattices of opens as a primitive notion rather than a set of points

In mathematics, pointless topology, also called point-free topology (or pointfree topology) or topology without points and locale theory, is an approach to topology where lattices of open sets are the primitive notion, and are not required to consist of subsets of points. In this approach it becomes possible to construct topologically interesting spaces from purely algebraic data.

Points are then a derived notion rather than primitive, and there are non-trivial spaces that have no points. 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). Karl Menger was an early pioneer in the field, and his work on topology without points was inspired by Whitehead's point-free geometry and used shrinking regions of the plane to simulate points. Apart from Stone, Henry Wallman also exploited this idea. Others continued this path till Charles Ehresmann and his student Jean Bénabou (and simultaneously others), took a major step in the late fifties. Their insights arose from the study of "topological" and "differentiable" categories. Ehresmann's approach involved using a category whose objects were complete lattices that satisfied a distributive law and whose morphisms were maps that preserved finite meets and arbitrary joins.

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