Abstract
We develop the first steps of a constructive theory of uniformities given by pseudometrics and study its relation to the constructive theory of Bishop topologies. Both these concepts are constructive, function-theoretic alternatives to the notion of a topology of open sets. After motivating the constructive study of uniformities of pseudometrics we present their basic theory and we prove a Stone-Cech theorem for them. We introduce the f-uniform spaces and we prove a Tychonoff embedding theorem for them. We study the uniformity of pseudometrics generated by some Bishop topology and the pseudo-compact Bishop topology generated by some uniformity of pseudometrics. Defining the large uniformity on reals we prove a ``large'' version of the Tychonoff embedding theorem for f-uniform spaces and we show that the notion of morphism between uniform spaces captures Bishop continuity. We work within Bishop's informal system of constructive mathematics BISH extended with inductive definitions with rules of countably many premisses.
Highlights
We develop the first steps of a constructive theory of uniformities given by pseudometrics and study its relation to the constructive theory of Bishop topologies
In [21, page 216] Gillman and Jerison remark the following: From our point of view, the most efficient approach to uniform spaces is by way of pseudometrics, as they provide us with a large supply of continuous functions we define a uniform structure to be a family of pseudometrics
The concept of a locally convex space would appear to be important for constructive mathematics, since examples exist in profusion
Summary
A uniformity of pseudometrics was the first notion of uniformity, which was introduced by Weil in [47] as a natural generalization of the notion of a metric. Rather it forces one to find a notion of compact uniformity of pseudometrics that does not copy the definition of a compact metric and at the same time is reduced to it when the uniform space is a metric one Such an enterprise with respect to compactness has been shown fruitful in formal topology (see Palmgren [30]), and in the theory of Bishop spaces (see Petrakis [36]). > 0 such that {(x, y) ∈ X × X | ∀1≤j≤n(dj(x, y) ≤ )} ⊆ U, implies the weak limited principle of omniscience in the form ∀a,b∈R(a = b ∨ ¬(a = b)) This fact cannot be considered as an argument against the development of the constructive theory of uniformities of pseudometrics, since the aim of such a theory is not to capture all classical results governing the relation between uniformities of entourages and uniformities of pseudometrics, a relation which is based on the fact that classically set-theoretic and function-theoretic objects are treated . We describe the computational meaning of the theory of uniformities of pseudometrics (and of Bishop topologies) within the informal system BISH∗
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