The continuum is here presented as a formal space by means of a finitary inductive definition. In this setting a constructive proof of the Heine-Borel covering theorem is given. 1 Introduction It is well known that the usual classical proofs of the Heine-Borel covering theorem are not acceptable from a constructive point of view (cf. [vS, F]). An intuitionistic alternative proof that relies on the fan theorem was given by Brouwer (cf. [B, H]). In view of the relevance of constructive mathematics for computer science, relying on the connection between constructive proofs and computations, it is natural to look for a completely constructive proof of the theorem in its most general form, namely for intervals with real-valued endpoints. By using formal topology the continuum, as well as the closed intervals of the real line, can be defined by means of finitary inductive definitions. This approach allows a proof of the Heine-Borel theorem that, besides being constructive, can also be compl...

We give the basic deønitions for pointfree functional analysis and present constructive proofs of the Alaoglu and Hahn-Banach theorems in the setting of formal topology. 1 Introduction We present the basic concepts and deønitions needed in a pointfree approach to functional analysis via formal topology. Our main results are the constructive proofs of localic formulations of the Alaoglu and Helly-Hahn-Banach 1 theorems. Earlier pointfree formulations of the Hahn-Banach theorem, in a topostheoretic setting, were presented by Mulvey and Pelletier (1987,1991) and by Vermeulen (1986). A constructive proof based on points was given by Bishop (1967). In the formulation of his proof, the norm of the linear functional is preserved to an arbitrary degree by the extension and a counterexample shows that the norm, in general, is not preserved exactly. As usual in pointfree topology, our guideline is to deøne the objects under analysis as formal points of a suitable formal space. After this has...

The first paper in this thesis presents a machine checked formalisation, in Martin-Löf's type theory, of pointfree topology with applications to domain theory. In the other papers pointfree topology is used in an approach to constructive analysis. The continuum is defined as a formal space from a base of rational intervals. Then the closed rational interval [a, b] is defined as a formal space, in terms of the continuum, and the Heine-Borel covering theorem is proved constructively. The basic definitions for a pointfree approach to functional analysis are given in such a way that the linear functionals from a seminormed linear space to the reals are points of a particular formal space, and in this setting the Alaoglu and the Hahn-Banach theorems are proved in an entirely constructive way. The proofs have been carried out in intensional Martin-Löf type theory with one universe and finitary inductive definitions, and the proofs have also been mechanically checked in an implementation of that system. ...

this paper that if a formal topology S is set-presented, in the sense of Aczel, and has only maximal points, then the collection of its points Pt(S) is isomorphic to a set. This generalises a result by Curi (2001) on setrepresentability. The crucial result (Theorem 4.3) is that any suitably small subset of a point can be extended to a point picked from a prescribed small power set. This power set depends only on the data of the formal space. The proof involves what seems to be a new choice principle (Theorem 4.2). It is a generalisation of dependent choice from natural numbers to W-types and is provable in type theory