Pointless topology

Summary

Pointless topology studies spaces through their lattices of open sets instead of through their points. Its basic objects are frames, complete lattices in which finite meets distribute over arbitrary joins, and locales, which are frames viewed as generalized spaces. Every topological space gives a locale, its frame of open sets, and for sober spaces no information is lost. Some theorems that need the axiom of choice for spaces, such as the product theorem for compact spaces, hold for locales without it.