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.
History 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. 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.
Intuition 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 sets is again open. In pointless topology we take these properties of the lattice as fundamental, without requiring that the lattice elements be sets of points of some underlying space and that the lattice operation be intersection and union. Rather, point-free topology is based on the concept of a "realistic spot" instead of a point without extent. These "spots" can be joined (symbol ∨ {\displaystyle \vee } ), akin to a union, and we also have a meet operation for spots (symbol ∧ {\displaystyle \land } ), akin to an intersection. Using these two operations, the spots form a complete lattice. If a spot meets a join of others it has to meet some of the constituents, which, roughly speaking, leads to the distributive law
b ∧ ( ⋁ i ∈ I a i ) = ⋁ i ∈ I ( b ∧ a i ) {\displaystyle b\wedge \left(\bigvee _{i\in I}a_{i}\right)=\bigvee _{i\in I}\left(b\wedge a_{i}\right)}
where the a i {\displaystyle a_{i}} and b {\displaystyle b} are spots and the index family I {\displaystyle I} can be arbitrarily large. This distributive law is also satisfied by the lattice of open sets of a topological space. If X {\displaystyle X} and Y {\displaystyle Y} are topological spaces with lattices of open sets denoted by Ω ( X ) {\displaystyle \Omega (X)} and Ω ( Y ) {\displaystyle \Omega (Y)} , respectively, and f : X → Y {\displaystyle f\colon X\to Y} is a continuous map, then, since the pre-image of an open set under a continuous map is open, we obtain a map of lattices in the opposite direction: f ∗ : Ω ( Y ) → Ω ( X ) {\displaystyle f^{*}\colon \Omega (Y)\to \Omega (X)} . Such "opposite-direction" lattice maps thus serve as the proper generalization of continuous maps in the point-free setting.
Formal definitions The basic concept is that of a frame, a complete lattice satisfying the general distributive law:
b ∧ ( ⋁ i ∈ I a i ) = ⋁ i ∈ I ( b ∧ a i ) {\displaystyle b\wedge \left(\bigvee _{i\in I}a_{i}\right)=\bigvee _{i\in I}\left(b\wedge a_{i}\right)}
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