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Contributions to Pointfree Topology and Apartness Spaces UPPSALA DISSERTATIONS IN MATHEMATICS 71 Contributions to Pointfree Topology and Apartness Spaces Anton Hedin Department of Mathematics Uppsala University UPPSALA 2011 Dissertation presented at Uppsala University to be publicly examined in Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Wednesday, June 8, 2011 at 10:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Abstract Hedin, A. 2011. Contributions to Pointfree Topology and Apartness Spaces. Department of Mathematics. Uppsala Dissertations in Mathematics 71. 40 pp. Uppsala. ISBN 978-91-506-2219-5. The work in this thesis contains some contributions to constructive point-free topology and the theory of apartness spaces. The first two papers deal with constructive domain theory using formal topology. In Paper I we focus on the notion of a domain representation of a formal space as a way to introduce generalized points of the represented space, whereas we in Paper II give a constructive and point-free treatment of the domain theoretic approach to differential calculus. The last two papers are of a slightly different nature but still concern constructive topology. In paper III we consider a measure theoretic covering theorem from various constructive angles in both point-set and point-free topology. We prove a point-free version of the theorem. In Paper IV we deal with issues of impredicativity in the theory of apartness spaces. We introduce a notion of set-presented apartness relation which enables a predicative treatment of basic constructions of point-set apartness spaces. Keywords: Constructive mathematics, General topology, Pointfree topology, Domain theory, Interval analysis, Apartness spaces Anton Hedin, Department of Mathematics, Algebra, Geometry and Logic, Box 480, Uppsala University, SE-75106 Uppsala, Sweden. © Anton Hedin 2011 ISSN 1401-2049 ISBN 978-91-506-2219-5 urn:nbn:se:uu:diva-152068 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-152068) Till mamma och pappa List of Papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I Hedin, A. (2011) Local Scott compactification. Manuscript (submitted). II Hedin, A. (2011) The domain theoretic derivative in formal topology. Manuscript (submitted). III Diener, H., Hedin, A. (2011) The Vitali covering theorem in construc- tive mathematics. Manuscript (submitted). IV Hedin, A. (2011) Towards set-presentable apartness spaces. Manuscript. Contents 1 Introduction . 9 1.1 Constructive mathematics . 9 1.1.1 Intuitionistic mathematics . 10 1.1.2 Russian constructive mathematics . 12 1.1.3 Bishop’s constructive mathematics . 13 1.1.4 Sets and predicativity . 14 1.2 Constructive topology . 15 1.2.1 Point-free topology . 16 1.2.2 Formal spaces . 18 1.2.3 Apartness spaces . 22 2 Summary of papers . 25 2.1 Paper I . 25 2.2 Paper II . 27 2.3 Paper III . 28 2.4 Paper IV . 30 3 Sammanfattning på svenska (Summary in Swedish) . 33 4 Acknowledgements . 35 Bibliography . 37 1. Introduction The work in this thesis contains some contributions to constructive point-free topology and the theory of apartness spaces. The first two papers deal with constructive domain theory using formal topology. Paper I focuses on the no- tion of a domain representation of a formal space as a way to introduce gen- eralized points of the represented space, whereas Paper II gives a constructive treatment of the domain theoretic approach to differential calculus. The last two papers are of a different nature but still concern constructive topology. In Paper III we consider a measure theoretic covering theorem from various constructive angles in both point-set and point-free topology. Paper IV deals with issues of impredicativity in the theory of apartness spaces. By constructive we mean that we work in Bishop style constructive math- ematics, that is mathematics based on intuitionistic logic together with a suit- able theory of sets. We will consider two other variants of constructivism in Paper III. This introduction serves to give a very brief background to, as well as introduce some important concepts of, constructive mathematics in general and constructive topology in particular. For more background on constructive mathematics and its history see [4, 8, 45, 46]. 1.1 Constructive mathematics The traditional mathematical practice, which is usually referred to as classical mathematics, with its two-valued interpretation of mathematical statements as truth values, favors the conception that mathematical objects exist independent of human activity. In this ideal universe every meaningful statement about its inhabitants has a definite answer. Without having a proof of either of two contradicting statements the classical mathematician can always assert that one of them must be true. This is the Law of Excluded Middle (LEM): for every statement P, either P or :P. In the beginning of the 20th century L.E.J. Brouwer, famous Dutch topolo- gist (1881-1966), challenged the prevailing mathematical culture by develop- ing mathematics according to the view that mathematical objects are the cre- ations of the human mind and every proof of a mathematical theorem must be accompanied by a so called mental construction. Brouwer’s approach forced a reinterpretation of the logical connectives and quantifiers. For example a (mental) construction for a disjunctive statement "P or Q", in logical symbols 9 P _ Q, is given by either a construction for P or a construction for Q and the information which of P and Q the construction belongs to. In turn this leads to the immediate rejection of the principle LEM. This interpretation was later clarified by one of Brouwer’s students A. Heyting, who explained the meaning of a statement in terms of what counts as a constructive proof of the statement. • A proof of P ^ Q consists of a proof of P and a proof of Q. • A proof of P _ Q consists of a proof of P or a proof of Q, together with the information which disjunct is proven. • A proof of P ! Q is a construction which transforms any proof of P into a proof of Q. • Absurdity ? (contradiction) has no proof. Negations :P are defined as P !?. • A proof of (8x)P(x) is a construction which transforms an arbitrary individual a into a proof of P(a). • A proof of (9x)P(x) consists of an individual a and a proof of P(a). A similar interpretation was given independently by A.N. Kolmogorov, in terms of problems and solutions, and therefore it is commonly referred to as the Brouwer-Heyting-Kolmogorov interpretation, or BHK interpretation. The interesting clauses are the ones explaining proofs of disjunctive and existential statements. There is a clear distinction between the construction of a mathe- matical object and an indirect proof of its existence, which is sufficient in classical mathematics. To say that there is an x such that P(x) holds means that we have some explicit way of constructing an x for which we can prove that P(x) holds. A proof that the assumption that no x satisfies P(x) leads to a contradiction, does not count as such a construction. The BHK interpretation is indeed quite informal as it depends on the mean- ing of the notion of construction. The exact understanding of this notion dif- fers between various schools of constructive mathematics. We will briefly describe three schools of constructivism: intuitionistic mathematics, Russian constructive mathematics and Bishop’s constructive mathematics. The latter two interpret the notion of construction as algorithm, with the distinction that the former provides a strict formalism for defining algorithms as well as treating the algorithms themselves as mathematical objects, whereas the latter takes the notion of algorithm as fundamental. 1.1.1 Intuitionistic mathematics This is the mathematical practice of Brouwer and his followers. A distinctive feature is its treatment of choice sequences a 2 NN of natural numbers. Based 10 on the idea of the (ideal) mathematician as the creating subject, a sequence of natural numbers can be constructed over time by a free choice at every time step. An intuitive example is given by the sequence of consecutive tosses of an (abstract) die. Such a sequence is to be contrasted with the so called law-like sequences, which are completely determined by a law or rule, e.g. the constant sequence (lx:0). In this sense the construction of an arbitrary sequence a 2 NN is always incomplete and we can in general only know, at any stage, an initial segment an := ha(0);:::;a(n − 1)i of the sequence. A function f assigning a number f (a) 2 N to every choice sequence a can thus also only depend on some initial segment am, and hence we must have f (b) = f (a) for all sequences b 2 NN with the same initial segment bm = am. Thus one is led to accept the principle of continuous choice (CC) which consists of a continuity part, explained by the previous argument, and a choice part, following from the BHK interpretation CC1. Every function from NN to N is continuous CC2. If P ⊆ NN × N, and for every a 2 NN there exists n 2 N such that (a;n) 2 P, then there exists a function f : NN ! N such that (a; f (a)) 2 P for every a 2 NN. A consequence of the continuity principle is that every function f : R ! R is continuous. In classical mathematics the continuity principle is simply false because of CC1. A fundamental part of intuitionistic mathematics (INT) is the fan theorem (FT). Before we can state it we need to define some notions, which will appear again in Paper III. For a set A, we denote by A∗ the set of all finite sequences ∗ of elements of A.
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