Pointless topology

The pointless topology is an approach to topology that avoids mentioning points. A traditional topological space consists of a set of "points", together with a set of "open sets". These open sets form a lattice with certain properties. Pointless topology studies the lattices themselves as abstract entities, without reference to an underlying set of points. Since some of the lattices thus defined do not come from topological spaces, one can see the category of pointless topological spaces, also called locals, as an extension of the category of ordinary topological spaces. Some authors claim that this new category has certain natural properties that make it preferable. Details on the relationship between the category of topological spaces and the category of locals, including the explicit construction of the duality between sober spaces and spatial locales, can be found in Stone's article on duality.

Formally, we define a frame as a lattice L in which each (still infinite) subset {a_i} has a supremum Va_i such that (full distribution)

b ∧ (V a_i) = V (a_i ∧ b)

for every b and every set {a_i} of L. These frames, together with the lattice homomorphisms that respect arbitrary suprema, form a category; the opposite category of the category of frames is called the category of locals and generalizes the category of topological spaces. The reason we take the opposite category is that every continuous function f: X → Y between the topological spaces induces a function between the lattices of open sets in the opposite direction: each open set O in Y is mapped to the open set f ^--1(O) in X.

It is possible to translate most of the concepts of point topology into the context of locales, and prove the analogous theorems. While many important theorems in point topology require the axiom of choice, this is not true of their analogues in local theory. This can be useful if one works on a topos that does not satisfy the axiom of choice. The concept of "local product" it diverges slightly from the "product of topological spaces" concept, and this divergence has been considered a disadvantage of the locals approach. Others affirm that the local product is more natural and point to several of its "desirable" which are not shared by the products of topological spaces.

See also Heyting algebra. A local is a complete Heyting algebra.