[1] In this approach it becomes possible to construct topologically interesting spaces from purely algebraic data.
[2] 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" categories.
[2] 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.
[3] 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.
[4] 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.
Rather, point-free topology is based on the concept of a "realistic spot" instead of a point without extent.
), akin to a union, and we also have a meet operation for spots (symbol
This distributive law is also satisfied by the lattice of open sets of a topological space.
are topological spaces with lattices of open sets denoted by
The basic concept is that of a frame, a complete lattice satisfying the general distributive law above.
Frame homomorphisms are maps between frames that respect all joins (in particular, the least element of the lattice) and finite meets (in particular, the greatest element of the lattice).
If we restrict this functor to the full subcategory of sober spaces, we obtain a full embedding of the category of sober spaces and continuous maps into the category of locales.
It is possible to translate most concepts of point-set topology into the context of locales, and prove analogous theorems.
Some important facts of classical topology depending on choice principles become choice-free (that is, constructive, which is, in particular, appealing for computer science).
Thus for instance, arbitrary products of compact locales are compact constructively (this is Tychonoff's theorem in point-set topology), or completions of uniform locales are constructive.
Another point where topology and locale theory diverge strongly is the concepts of subspaces versus sublocales, and density: given any collection of dense sublocales of a locale
[6] This leads to Isbell's density theorem: every locale has a smallest dense sublocale.
A general introduction to pointless topology is This is, in its own words, to be read as a trailer for Johnstone's monograph and which can be used for basic reference: There is a recent monograph For relations with logic: For a more concise account see the respective chapters in: