Editing Pointless topology (section)

==Intuition==
Traditionally, a [[topological space]] consists of a [[Set (mathematics)|set]] of [[point (topology)|points]] together with a ''topology'', a system of subsets called [[open set]]s that with the operations of  [[union (set theory)|union]] (as [[Join (mathematics)|join]]) and [[intersection (set theory)|intersection]] (as  [[Meet (Mathematics)|meet]]) forms a [[lattice (order)|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 [[Join (mathematics)|joined]] (symbol <math>\vee </math>), akin to a union, and we also have a [[Meet (Mathematics)|meet]] operation for spots (symbol <math>\and </math>), 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

:<math>b \wedge \left( \bigvee_{i\in I} a_i\right) = \bigvee_{i\in I} \left(b\wedge a_i\right)</math>
where the <math>a_i</math> and <math>b</math> are spots and the index family <math>I</math> can be arbitrarily large. This distributive law is also satisfied by the lattice of open sets of a topological space.

If <math>X</math> and <math>Y</math> are topological spaces with lattices of open sets denoted by <math>\Omega(X)</math> and <math>\Omega(Y)</math>, respectively, and <math>f\colon X\to Y</math> is a [[Continuous function|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: <math>f^*\colon \Omega(Y)\to \Omega(X)</math>. Such "opposite-direction" lattice maps thus serve as the proper generalization of continuous maps in the point-free setting.