In this paper we introduce relative formal topology so that we can deal constructively with the closed subsets of a topological space as well as with the open ones. In fact, within standard formal topology, the cover relation allows the definition of the formal opens, which are supposed to act as the formal counterparts of the open subsets, and within balanced formal topology, the binary positivity predicate allows the definition of the formal closed subsets, which are supposed to act as the formal counterparts of the closed subsets. However, these approaches are only fully satisfactory (according to the criterion introduced by the author in Valentini (2005)) if we can provide an adequate formalisation of the cover relation and the positivity predicate. But current formalisations fail in this respect since some intuitionistically valid relations between the open and closed subsets of a concrete topological space cannot be expressed. Our central result is that we can solve this problem through a generalisation of the standard cover relation together with a binary positivity predicate satisfying the positivity axiom.

Abstract By using some classical reasoning we show that any countably presented formal topology, namely, a formal topology with a countable axiom set, is spatial.

Abstract. We present and study the category of formal topologies and some of its variants. Two main results are proven. The first is that, for any inductively generated formal cover, there exists a formal topology whose cover extends in the minimal way the given one. This result is obtained by enhancing the method for the inductive generation of the cover relation by adding a coinductive generation of the positivity predicate. Categorically, this result can be rephrased by saying that inductively generated formal topologies are coreflective into inductively generated formal covers. The second result is that unary formal covers are exponentiable in the category of inductively generated formal covers and hence, thanks to the coreflection, unary formal topologies are exponentiable in the category of inductively generated formal topologies. From a localic point of view the exponentiability of unary formal topologies means that algebraic dcpos are exponentiable in the category of open locales. But, the coreflection theorem states that open locales are coreflective in locales and hence, as a consequence of well-known impredicative results on exponentiable locales, it allows to prove that locally compact open locales are exponentiable in the category of open locales.

In this paper we give a constructive proof of the pointfree version of Tychonoff's theorem within formal topology, using ideas from Coquand's proof in [7]. To deal with pointfree topology Coquand uses Johnstone's coverages. Because of the representation theorem in [3], from a mathematical viewpoint these structures are equivalent to formal topologies but there is an essential difference also. Namely, formal topologies have been developed within Martin Löf's constructive type theory (cf. [16]), which thus gives a direct way of formalizing them (cf. [4]). The most important aspect of our proof is that it is based on an inductive definition of the topological product of formal topologies. This fact allows us to transform Coquand's proof into a proof by structural induction on the last rule applied in a derivation of a cover. The inductive generation of a cover, together with a modification of the inductive property proposed by Coquand, makes it possible to formulate our proof of Tychonoff s theorem in constructive type theory. There is thus a clear difference to earlier localic proofs of Tychonoff's theorem known in the literature (cf. [9, 10, 12, 14, 27]). Indeed we not only avoid to use the axiom of choice, but reach constructiveness in a very strong sense. Namely, our proof of Tychonoff's theorem supplies an algorithm which, given a cover of the product space, computes a finite subcover, provided that there exists a similar algorithm for each component space.