Pointlessly - Relation To Point-set Topology

Relation To Point-set Topology

It is possible to translate most concepts of point-set topology into the context of locales, and prove analogous theorems. While many important theorems in point-set topology require the axiom of choice, this is not true for some of their analogues in locale theory. This can be useful if one works in a topos that does not have the axiom of choice.

The concept of "product of locales" diverges slightly from the concept of "product of topological spaces", and this divergence has been called a disadvantage of the locale approach. Others claim that the locale product is more natural, and point to several "desirable" properties not shared by products of topological spaces.

For almost all spaces (more precisely for sober spaces), the topological product and the localic product have the same set of points. The products differ in how equality between sets of open rectangles, the canonical base for the product topology, is defined: equality for the topological product means the same set of points is covered; equality for the localic product means provable equality using the frame axioms. As a result, two open sublocales of a localic product may contain exactly the same points without being equal.

A point where locale theory and topology diverge much more strongly is the concept of subspaces vs. sublocales. The rational numbers have c subspaces but 2c sublocales. The proof for the latter statement is due to John Isbell, and uses the fact that the rational numbers have c many pairwise almost disjoint (= finite intersection) closed subspaces.