Pointless topology
In mathematics, pointless topology (also called point-free or pointfree topology, or locale theory) is an approach to topology that avoids mentioning points. The name 'pointless topology' is due to John von Neumann. The ideas of pointless topology are closely related to mereotopologies in which regions (sets) are treated as foundational without explicit reference to underlying point sets.
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 intersection and union forms a lattice with certain properties. Pointless topology then studies lattices like these abstractly, without reference to any underlying set of points. Since some of the so-defined lattices do not arise from topological spaces, one may see the category of pointless topological spaces, also called locales, as an extension of the category of ordinary topological spaces.
Formally, a frame is defined to be a lattice L in which finite meets distribute over arbitrary joins, i.e. every (even infinite) subset {ai} of L has a supremum ⋁ai such that
for all b in L. These frames, together with lattice homomorphisms that respect arbitrary suprema, form a category. The dual of the category of frames is called the category of locales and generalizes the category Top of all topological spaces with continuous functions. The consideration of the dual category is motivated by the fact that every continuous map between topological spaces X and Y induces a map between the lattices of open sets in the opposite direction as for every continuous function f: X → Y and every open set O in Y the inverse image f -1(O) is an open set in X.
...
Wikipedia