It is a kind of pointless topology; in the context of classical mathematics, it reproduces the theory of locales rather than topological spaces (although of course one can recover topological spaces from locales).
The basic definitions can be motivated by an attempt to study locales entirely through the posites that generate them. However, in order to recover all basic topological notions (particularly those associated with closed rather than open features) predicatively, we need to add a ‘positivity’ relation to the ‘coverage’ relation of sites.
A formal topology or formal space is a set together with
for all , , , and .
We interpret the elements of as basic opens in the formal space. We call the entire space and and the intersection of and . We say that is covered by or that is a cover of if . We say that is positive or inhabited if . (For a topological space equipped with a strict topological base , taking these intepretations literally does in fact define a formal space; see the Examples.)
Some immediate points to notice: * If we drop (1), then the hypothesis of (1) defines an equivalence relation on which is a congruence for , , , and , so that we may simply pass to the quotient set. In appropriate foundations, we can even allow to be a preset originally, then use (1) as a definition of equality. * We can prove that is a bounded semilattice; if (as the notation suggests) we interpret this as a meet-semilattice, then if and only if . Conversely, we could require that be a semilattice originally, then let (1) say that whenever . * We can prove that holds iff every cover of is inhabited and that fails iff . Accordingly, this predicate is uniquely definable (in two equivalent ways, one impredicative and one nonconstructive) in a classical treatment; only in a treatment that is both predicative and constructive do we need to include it in the axioms. See positivity predicate.
Let be a topological space, and let be the collection of open subsets of . Let be itself, and let be the literal intersection of and for . Let if and only is literally an open cover of , and let if and only if is literally inhabited. Then is a formal topology.
The above example is impredicative (since the collection of open subsets is generally large), but now let be a base for the topology of which is strict in the sense that it is closed under finitary intersection. Let the other definitions be as before. Then is a formal topology.
More generally, let be a subbase for the topology of , and let be the free monoid on , that is the set of finite lists of elements of (so this example is not strictly finitist), modulo the equivalence relation by which two lists are identified if their intersections are equal. Let be the empty list, let be the concatenation of and , let if the intersection of is contained in the union of the intersections of the elements of , and let if the intersection of is inhabited. Then is a formal topology.
Let be an accessible locale generated by a posite whose underlying poset is a (meet)-semilattice. Let and be as in the semilattice structure on , and let if contains a basic cover (in the posite structure on ) of . Then we get a formal topology, defining in the unique way.
The last example is not predicative, and this is in part why one studies formal topologies instead of sites, if one wishes to be strictly predicative. (It still needs to be motivated that we want at all.)
Mike Fourman and Grayson (1982); Formal Spaces. This is the original development, intended as an application of locale theory to logic.