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A green link indicates that the item is available online at least partially. This experiment has been authorized by the editors of the Stanford Encyclopedia of Philosophy. The original article and bibliography can be found here. We derive similar and new results on the efficient representation of continuous real-valued functions defined on a suitable topological space. This can be generalized to continuous functions between suitable topological spaces.

It combines a pointfree with a pointwise approach, by integrating the partial-order on the topological basis with a pre-apartness relation. Simpler than formal topology and constructive domain theory, NToP enables a smooth transition from theory to practice.

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The construction directly yields an apartness topology which is equivalent to the metric topology. This allows for an efficient representation of compact spaces by finitely branching trees contrasting to the framework of formal topology.

1. Introduction

For the natural reals, we can suffice with lean dyadic intervals. All our spaces are also pointwise topological spaces, enabling the familiar pointwise style of BISH and CLASS, as well as an incorporation of earlier constructive work in analysis.

In the case of the reals, every BISH-continuous real function can thus be represented by a morphism sending lean dyadic intervals to lean dyadic intervals. This then should be enough validation also for BISH.

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Errata : So far I have found some minor typos, and an imprecise remark, which I will correct shortly. Natural topology is tailored to make pointwise and pointfree notions go together naturally. As a constructive theory in BISH, it gives a classical mathematician a faithful idea of important concepts and results in intuitionism. Natural topology is well-suited for practical and computational purposes. We give several examples relevant for applied mathematics, such as the decision-support system Hawk-Eye, and various real-number representations.


To do so, our mathematical foundation must be precise and simple. There are links with physics, regarding the topological character of our physical universe. Any natural space is isomorphic to a quotient space of Baire space, which therefore is universal. Inductive morphisms respect this Heine-Borel property, inversely. This partly solves the continuous-function problem for BISH, yet pointwise problems persist in the absence of Brouwer's Thesis.

Constructive mathematics and models of type theory

By inductivizing the definitions, a direct correspondence with INT is obtained which allows for a translation of many intuitionistic results into BISH. We thus prove a constructive star-finitary metrization theorem which parallels the classical metrization theorem for strongly paracompact spaces. We also obtain non-metrizable Silva spaces, in infinite-dimensional topology.

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Natural topology gives a solid basis, we think, for further constructive study of topological lattice theory, algebraic topology and infinite-dimensional topology. Verification that certain procedures should work under appropriate conditions will give students good examples of the application of real analysis and implementation will require them to be able to make and run simple procedures in Matlab. The Division Algorithm on Integers, Euclid's Algorithm including proof of termination with highest common factor.

The solution of simple linear Diophantine equations. Root finding for real polynomials. Fixed point iterations, examples.

intuitionistic topology & foundations of constructive mathematics

Existence of fixed points and convergence of fixed point iterations by the contraction mapping theorem using the mean value theorem. Skip to main content. Mathematical Institute Course Management.