Heine-Borel Research Articles

On the logical and computational properties of the Vitali covering theorem

We study a version of the Vitali covering theorem, which we call WHBU and which is a direct weakening of the Heine-Borel theorem for uncountable coverings, called HBU. We show that WHBU is central to measure theory by deriving it from various central approximation results related to Littlewood's three principles. A natural question is then how hard it is to prove WHBU (in the sense of Kohlenbach's higher-order Reverse Mathematics), and how hard it is to compute the objects claimed to exist by WHBU (in the sense of Kleene's computation schemes S1-S9). The answer to both questions is ‘extremely hard’, as follows: on one hand, in terms of the usual scale of (conventional) comprehension axioms, WHBU is only provable using Kleene's ∃3, which implies full second-order arithmetic. On the other hand, realisers (aka witnessing functionals) for WHBU, so-called Λ-functionals, are computable from Kleene's ∃3, but not from weaker comprehension functionals. Despite this hardness, we show that WHBU, and certain Λ-functionals, behave much better than HBU and the associated class of realisers, called Θ-functionals. In particular, we identify a specific Λ-functional called ΛS which adds no computational power to the Suslin functional, in contrast to Θ-functionals. Finally, we introduce a hierarchy involving Θ-functionals and HBU.

The Hyperreals enlargement sets and its application on compact topology

This paper presents a study of hyperreal numbers and their relations to real numbers. The hyperreals are a number system extension of the real number system. With this system, simpler and more intuitively natural definition (topological notions of special points and compactness), proofs (Heine-Borel theorem), and new concepts (limited, unlimited numbers, enlargement sets, and halos) of mathematical interest are offered.

On Some Aspects of Compactness in Metric Spaces

In this paper, we investigate the generalizations of the concepts from Heine-Borel Theorem and the Bolzano-Weierstrass Theorem to metric spaces. We show that the metric space X is compact if every open covering has a finite subcovering. This abstracts the Heine-Borel property. Indeed, the Heine-Borel Theorem states that closed bounded subsets of the real line R are compact. In this study, we rephrase compactness in terms of closed bounded subsets of the real line R, that is, the Bolzano-Weierstrass theorem. Let X be any closed bounded subset of the real line. Then any sequence (xn) of the points of X has a subsequence converging to a point of X. We have used these interesting theorems to characterize compactness in metric spaces.

Metric completions, the Heine-Borel property, and approachability

Abstract We show that the metric universal cover of a plane with a puncture yields an example of a nonstandard hull properly containing the metric completion of a metric space. As mentioned by Do Carmo, a nonextendible Riemannian manifold can be noncomplete, but in the broader category of metric spaces it becomes extendible. We give a short proof of a characterisation of the Heine-Borel property of the metric completion of a metric space M in terms of the absence of inapproachable finite points in ∗ M.

Azela-Ascoli Theorem and Its Applications

In this study we looked at the theorm Azela-Ascoli and its application to functional analysis, ordinary dierential equations and complex analysis. This theorem simplies the checking of compactness for subsets of spaces of continuous functions in much the same way the Heine-Borel theorem does for subsets of Rn

From pairwise comparisons to consistency with respect to a group operation and Koczkodaj's metric

Important in pairwise comparisons methods (PC) concepts of reciprocal matrices, consistency conditions and priority vectors of consistent PC matrices are investigated in a more general framework where mappings take values in a not necessarily abelian group. A problem about Pauli's matrices is posed. Notions of left and right priority mappings with respect to a group operation of consistent mappings are introduced. Hou's fixed point theorems on consistent matrices are generalized to theorems on consistent mappings with values in an arbitrarily fixed lattice-ordered group. Relevant to topology basic properties of Koczkodaj's metric which induces Koczkodaj's inconsistency index in PC are studied. It is shown in ZF that Koczkodaj's metric is Cantor complete and not totally bounded, it is equivalent with the Euclidean metric on the set of positive real numbers. A version of Heine–Borel Theorem for Koczkodaj's metric is obtained and briefly applied to the compact bornology of R+n.

Nets and reverse mathematics

Nets are generalisations of sequences involving possibly uncountable index sets; this notion was introduced about a century ago by Moore and Smith. They also established the generalisation to nets of various basic theorems of analysis due to Bolzano–Weierstrass, Dini, Arzelà, and others. More recently, nets are central to the development of domain theory, providing intuitive definitions of the associated Scott and Lawson topologies, among others. This paper deals with the Reverse Mathematics study of basic theorems about nets. We restrict ourselves to nets indexed by subsets of Baire space, and therefore third-order arithmetic, as such nets suffice to obtain our main results. Over Kohlenbach’s base theory of higher-order Reverse Mathematics, the Bolzano–Weierstrass theorem for nets implies the Heine–Borel theorem for uncountable covers. We establish similar results for other basic theorems about nets and even some equivalences, e.g. for Dini’s theorem for nets. Finally, we show that replacing nets by sequences is hard, but that replacing sequences by nets can obviate the need for the Axiom of Choice, a foundational concern in domain theory. In an appendix, we study the power of more general index sets, establishing that the ‘size’ of a net is directly proportional to the power of the associated convergence theorem.

A transfer principle for second-order arithmetic, and applications

In the theory of conditional sets, many classical theorems from areas such as analysis, probability theory or measure theory are lifted to the conditional framework, often to be applied in areas such as mathematical economics or optimization. The frequent experience that such theorems can be proved by `conditionalizations' of the classical proofs suggests that a general transfer principle is in the background, and that formulating and proving such a transfer principle would yield a wealth of useful further conditional versions of classical results, in addition to providing a uniform approach to the results already known. In this paper, we formulate and prove such a transfer principle based on second-order arithmetic, which, by the results of reverse mathematics, suffices for the bulk of classical mathematics, including real analysis, measure theory and countable algebra, and excluding only more remote realms like category theory, set-theoretical topology or uncountable set theory, see e.g. the introduction of \cite{simpson2009subsystems}. This transfer principle is then employed to give short and easy proofs of conditional versions of central results in various areas of mathematics, including the Bolzano-Weierstrass theorem, the Heine-Borel theorem, the Riesz representation theorem and Brouwer's fixed point theorem.

A Short Proof of the Bolzano–Weierstrass Theorem

SummaryWe present a short proof of the Bolzano–Weierstrass theorem on the real line which avoids monotonic subsequences, Cantor's intersection theorem, and the Heine–Borel theorem.

Revisiting the construction of the real line

The most common constructions of the reals from the rationals attempt to “fill in the gaps,” using (equivalence classes of) Cauchy sequences of rationals or Dedekind cuts. Another technique, more familiar to logicians than to analysts, is to expand the set Q of rationals to the set QN of all sequences of rationals and then collapse back down in two stages. In the first stage, produce a structure X by identifying sequences that agree on a large set; in the second stage, further collapse X by removing its infinite elements, and collapsing its infinitesimals to 0. The latter approach, though less widely known, provides, in its intermediary construction X, an example of interesting pathologies. We show here that X is an example of a complete (even open-complete) ordered field that is not Archimedean. The field X showcases the significant role of the Archimedean property in the structure of R, as many of the fundamental properties of R are found to fail in X precisely because this property is absent, such as the Heine-Borel Theorem and the Bolzano-Weierstrass Theorem. Indeed, we show that the field X is not even metrizable. We conclude with a detailed proof, accessible to non-logicians, that the second stage of the construction succeeds in producing a complete Archimedean ordered field. Mathematics Subject Classification: 12J15, 26E35

Compact Operators on Hilbert Spaces

In this paper, we obtain some results on compact operators. More specially, we prove that if T is a unitary operator on a Hilbert space H, then it is compact if and only if H has a finite dimension. Also, we prove that, if H is a Hilbert space with Heine-Borel property, then K(H) = BL(H).

More reverse mathematics of the Heine-Borel Theorem

Using the techniques of reverse mathematics, we characterize subsets X (0; 1) in terms of the strength of HB(X), the Heine-Borel Theorem for the subset. We introduce W(X), formalizing the notion that the Heine-Borel Theorem for X is weak, and S(X), formalizing the notion that the theorem is strong. Using these, we can prove the following three results: RCA0' W(X)! HB(X), RCA0' S(X)! (HB(X)! WKL0), and ATR0' X exists! (W(X)_ S(X)). 2010 Mathematics Subject Classification 03B30, 03F35 (primary); 03F60, 26E40 (secondary)

Localic completion of generalized metric spaces II: Powerlocales

The work investigates the powerlocales (lower, upper, Vietoris) of localic completions of generalized metric spaces. The main result is that all three are localic completions of generalized metric powerspaces, on the Kuratowski finite powerset. This is a constructive, localic version of spatial results of Bonsangue et al. and of Edalat and Heckmann. As applications, a localic completion is always overt, and is compact iff its gener- alized metric space is totally bounded. The representation is used to discuss closed intervals of the reals, with the localic Heine-Borel Theorem as a consequence. The work is constructive in the topos-valid sense. 2000 Mathematics Subject Classification 54B20 (primary); 06D22, 03G30,54E50,03F60 (secondary)

The anti-Specker property, a Heine–Borel property, and uniform continuity

Working within Bishop’s constructive framework, we examine the connection between a weak version of the Heine–Borel property, a property antithetical to that in Specker’s theorem in recursive analysis, and the uniform continuity theorem for integer-valued functions. The paper is a contribution to the ongoing programme of constructive reverse mathematics.

Compactness and finite dimension in asymmetric normed linear spaces

We describe the compact sets of any asymmetric normed linear space. After that, we focus our attention in finite dimensional asymmetric normed linear spaces. In this case we establish the equivalence between T 1 separation axiom and normable spaces. It is proved an asymmetric version of the Riesz Theorem about the compactness of the unit ball. We also prove that the Heine–Borel Theorem characterizes finite dimensional asymmetric normed linear spaces that satisfies the T 2 separation axiom. Finally we focus our attention on the T 0 separation axiom and results that are related to the dual p-complexity spaces.

Equivalents of the (Weak) Fan Theorem

This article presents a weak system of intuitionistic second-order arithmetic, WKV, a subsystem of the one in S.C. Kleene, R.E. Vesley [The Foundations of Intuitionistic Mathematics: Especially in Relation to Recursive Functions, North-Holland Publishing Company, Amsterdam, 1965]. It is then shown that some statements of real analysis, like a version of the Heine–Borel Theorem, and some statements of logic, e.g. compactness of classical proposition calculus, are equivalent to the (Weak) Fan Theorem in this system.

On the Persistence of Particles

This paper is about the metaphysical debate whether objects persist over time by the selfsame object existing at different times (nowadays called `endurance' by metaphysicians), or by different temporal parts, or stages, existing at different times (called ` perdurance'). I aim to illuminate the debate by using some elementary kinematics and real analysis: resources which metaphysicians have, surprisingly, not availed themselves of. There are two main results, which are of interest to both endurantists and perdurantists. (1): I describe a precise formal equivalence between the way that the two metaphysical positions represent the motion of the objects of classical mechanics (both point-particles and continua). (2): I make precise, and prove a result about, the idea that the persistence of objects moving in a void is to be analysed in terms of tracking the continuous curves in spacetime that connect points occupied by matter. The result is entirely elementary: it is a corollary of the Heine-Borel theorem.

On the equivalence of the Heine-Borel and the Bolzano-Weierstrass theorems

The equivalence of the Heine-Borel theorem and the Bolzano-Weierstrass theorem is proved.

The Tarski–Kantorovitch prinicple and the theory of iterated function systems

We show how some results of the theory of iterated function systems can be derived from the Tarski–Kantorovitch fixed–point principle for maps on partialy ordered sets. In particular, this principle yields, without using the Hausdorff metric, the Hutchinson–Barnsley theorem with the only restriction that a metric space considered has the Heine–Borel property. As a by–product, we also obtain some new characterisations of continuity of maps on countably compact and sequential spaces.

Developments in Constructive Nonstandard Analysis

AbstractWe develop a constructive version of nonstandard analysis, extending Bishop's constructive analysis with infinitesimal methods. A full transfer principle and a strong idealisation principle are obtained by using a sheaf-theoretic construction due to I. Moerdijk. The construction is, in a precise sense, a reduced power with variable filter structure. We avoid the nonconstructive standard part map by the use of nonstandard hulls. This leads to an infinitesimal analysis which includes nonconstructive theorems such as the Heine–Borel theorem, the Cauchy–Peano existence theorem for ordinary differential equations and the exact intermediate-value theorem, while it at the same time provides constructive results for concrete statements. A nonstandard measure theory which is considerably simpler than that of Bishop and Cheng is developed within this context.