Topological spaces in fact lead to very special topoi called locales. The set of open subsets of a topological space determines a lattice. The axioms for a topological space cause these lattices to be complete Heyting algebras. The theory of locales takes this as its starting point. A locale is defined to be a complete Heyting algebra, and the elementary properties of topological spaces are re-expressed and reproved in these terms. The concept of a locale turns out to be more general than a topological space, in that every sober topological space determines a unique locale, but many interesting locales do not come from topological spaces. Because locales need not have points, the study of locales is somewhat jokingly called pointless topology.