Abstract
Abstract Stone Duality is a new paradigm for general topology in which computable continuous functions are described directly, without using set theory, infinitary lattice theory or a prior theory of discrete computation. Every expression in the calculus denotes both a continuous function and a program, and the reasoning looks remarkably like a sanitised form of that in classical topology. This is an introduction to ASD for the general mathematician, with application to elementary real analysis. This language is applied to the Intermediate Value Theorem: the solution of equations for continuous functions on the real line. As is well known from both numerical and constructive considerations, the equation cannot be solved if the function hovers near 0, whilst tangential solutions will never be found. In ASD, both of these failures, and the general method of finding solutions of the equation when they exist, are explained by the new concept of overtness. The zeroes are captured, not as a set, but by higher-type modal operators. Unlike the Brouwer degree of a mapping, these are naturally defined and (Scott) continuous across singularities of a parametric equation. Expressing topology in terms of continuous functions rather than using sets of points leads to treatments of open and closed concepts that are very closely lattice- (or de Morgan-) dual, without the double negations that are found in intuitionistic approaches. In this, the dual of compactness is overtness. Whereas meets and joins in locale theory are asymmetrically finite and infinite, they have overt and compact indices in ASD. Overtness replaces metrical properties such as total boundedness, and cardinality conditions such as having a countable dense subset. It is also related to locatedness in constructive analysis and recursive enumerability in recursion theory.
Highlights
This language is applied to the Intermediate Value Theorem: the solution of equations for continuous functions on the real line
Expressing topology in terms of continuous functions rather than using sets of points leads to treatments of open and closed concepts that are very closely lattice(or de Morgan-) dual, without the double negations that are found in intuitionistic approaches
When we have described our new λ-calculus for general topology, we shall apply it to the intermediate value theorem, which solves equations that involve continuous functions R → R
Summary
When we have described our new λ-calculus for general topology, we shall apply it to the intermediate value theorem, which solves equations that involve continuous functions R → R. On examination, that these imply the extra property that constructivists require, which is very mild: it is satisfied by any example in which one might reasonably expect to be able to compute a zero These conditions are weaker forms of openness, whilst the “general” theorem is about continuous functions. Augustin-Louis Cauchy gave this proof in [9, Note III] These methods are not suitable as they stand for numerical solution of equations: Example 1.2 Consider this parametric function, which hovers around 0: for − 1 ≤ s ≤ +1 and 0 ≤ x ≤ 3, let fs(x) ≡ min x − 1, max (s, x − 2).
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