This note establishes that the familiar internal characterizations of the Tychonoff spaces whose rings of continuous real-valued functions are complete, or σ-complete, as lattice ordered rings already hold in the larger setting of pointfree topology. In addition, we prove the corresponding results for rings of integer-valued functions.

The familiar classical result that a continuous map from a space $X$ to a space $Y$ can be defined by giving continuous maps $varphi_U: U to Y$ on each member $U$ of an open cover ${mathfrak C}$ of $X$ such that $varphi_Umid U cap V = varphi_V mid U cap V$ for all $U,V in {mathfrak C}$ was recently shown to have an exact analogue in pointfree topology, and the same was done for the familiar cla...

The aim of this note is to show how various facts in classical topology connected with semicontinuous functions and the semicontinuous quasi-uniformity have their counterparts in pointfree topology. In particular, we introduce the localic semicontinuous quasi-uniformity, which generalizes the semicontinuous quasiuniformity of a topological space (known to be one of the most important examples o...

Properties of topological spaces are not absolute, as they may depend on the ambient mathematical universe. For instance, the answers to some questions in classical point-set topology hinge on non-standard set-theoretic axioms, while in intuitionistic mathematics even such basic properties as the Heine-Borel compactness of the closed interval depend on the ambient constructive variety. We show ...

The rst 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 de ned as a formal space from a base of rational intervals. Then the closed rational interval [a; b] is de ned as a formal space, in t...

We give the basic de nitions for pointfree functional analysis and present constructive proofs of the Alaoglu and Hahn Banach theorems in the set ting of formal topology Introduction We present the basic concepts and de nitions 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 H...

Partial frames provide a rich context in which to do pointfree structured and unstructured topology. A small collection of axioms of an elementary nature allows one to do much traditional pointfree topology, both on the level of frames or locales, and that of uniform or metric frames. These axioms are sufficiently general to include as examples bounded distributive...