Metric spaces in synthetic topology

We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and metric topologies of complete separable metric spaces coincide if they do so for Baire space. In Russian Constructivism the match between synthetic and metric topology breaks down, as even a very simple complete totally bounded space fails to be compact, and its topology is strictly finer than the metric topology. In contrast, in Brouwer’s intuitionism synthetic and metric notions of topology and compactness agree.     

Unique solutions

It is folklore that if a continuous function on a complete metric space has approximate roots and in a uniform manner at most one root, then it actually has a root, which of course is uniquely determined. Also in Bishop's constructive mathematics with countable choice, the general setting of the present note, there is a simple method to validate this heuristic principle. The unique solution even becomes a continuous function in the parameters by a mild modification of the uniqueness hypothesis. Moreover, Brouwer's fan theorem for decidable bars turns out to be equivalent to the statement that, for uniformly continuous functions on a compact metric space, the crucial uniform “at most one” condition follows from its non‐uniform counterpart. This classification in the spirit of the constructive reverse mathematics, as propagated by Ishihara and others, sharpens an earlier result obtained jointly with Berger and Bridges. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Intuitionistic sequential compactness?

Perhaps because the classical notion of sequential compactness fails to apply constructively even to $\{0,1\}$, Brouwer and his successors have paid little attention to the possibility of a constructive counterpart that is classically equivalent to sequential compactness and has serious potential for applications in analysis. We discuss such a notion – the anti-Specker property – and its equivalence, over Bishop-style constructive mathematics, to Brouwer’s fan theorem for $c$-bars.     

The anti‐Specker property,positivity, and total boundedness

Working within Bishop‐style constructive mathematics, we examine some of the consequences of the anti‐Specker property, known to be equivalent to a version of Brouwer's fan theorem. The work is a contribution to constructive reverse mathematics (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On weak compactness and countable weak compactness in fixed point theory

We prove that weak compactness and countable weak compactness in metric spaces are not equivalent. However, if the metric space has normal structure, they are equivalent. It follows that some fixed point theorems proved recently are consequences of a classical theorem of Kirk.

     

The Kripke schema in metric topology

A form of Kripke's schema turns out to be equivalent to each of the following two statements from metric topology: every open subspace of a separable metric space is separable; every open subset of a separable metric space is a countable union of open balls. Thus Kripke's schema serves as a point of reference for classifying theorems of classical mathematics within Bishop‐style constructive reverse mathematics.     

Generalising compactness

Working within the framework of Bishop's constructive mathematics, we will show that it is possible to define compactness in a more general setting than that of uniform spaces. It is also shown that it is not possible to do this in a topological space. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Kolmogorov–Riesz compactness theorem

We show that the Arzelà–Ascoli theorem and Kolmogorov compactness theorem both are consequences of a simple lemma on compactness in metric spaces. Their relation to Helly's theorem is discussed. The paper contains a detailed discussion on the historical background of the Kolmogorov compactness theorem.     

The Vietoris hyperspace structure for approach spaces

We show that there exists a natural approach version of the topological Vietoris hyperspace construction [16], [17] which has several advantages over the topological version. In the first place the important structural fact that the Vietoris construction can now also be considered, not only for topological but also intrinsically for metric spaces, but equally important in the second place the fact that we can considerably strengthen fundamental classic results. In this paper we mainly pay attention to properties concerning or involving compactness. As main results, in the first place we prove that it is not merely compactness of the Vietoris hyperspace which is equivalent to compactness of the original space [3] but that actually in the approach setting the indices of compactness [7], [8], [9], [10] numerically completely coincide. In the second place the well-known result [3], [4], [15] which says that if the original space is compact metric then the Vietoris topology is metrizable by the Hausdorff metric gets strengthened in the sense that in the approach setting under the same conditions the Vietoris approach structure actually coincides with the Hausdorff metric. Classic results follow as easy corollaries. Besides these main results we also draw attention to the good functorial relationship between the Vietoris approach structures and the associated topologies.     

Compactness for manifolds and integral currents with bounded diameter and volume

By Gromov??s compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance. Working in the class of oriented k-dimensional Riemannian manifolds (with boundary) and, more generally, integral currents in metric spaces in the sense of Ambrosio?CKirchheim and replacing the Hausdorff distance with the filling volume or flat distance, we prove an analogous compactness theorem in which however we only assume uniform bounds on volume and diameter.     

The swap of integral and limit in constructive mathematics

Integration within constructive, especially intuitionistic mathematics in the sense of L. E. J. Brouwer, slightly differs from formal integration theories: Some classical results, especially Lebesgue's dominated convergence theorem, have tobe substituted by appropriate alternatives. Although there exist sophisticated, but rather laborious proposals, e.g. by E. Bishop and D. S. Bridges (cf. [2]), the reference to partitions and the Riemann‐integral, also with regard to the results obtained by R. Henstock and J. Kurzweil (cf. [9], [12]), seems to give a better direction. Especially, convergence theorems can be proved by introducing the concept of “equi‐integrability”. The paper is strongly motivated by Brouwer's result that each function fully defined on a compact interval has necessarily to be uniformly continuous. Nevertheless, there are, with only one exception (a corollary of Theorem 4.2), no references to the fan‐theorem or to bar‐induction. Therefore, the whole paper can be read within the setting of Bishop's access to constructive mathematics. Nothing of genuine full‐fledged Brouwerian intuitionism is used for the main results in this note (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Compactness notions for an apartness space

Two new notions of compactness, each classically equivalent to the standard classical one of sequential compactness, for apartness spaces are examined within Bishop-style constructive mathematics.     

Continuous homomorphisms of R onto a compact group

It is shown within Bishop's constructive mathematics that, under one extra, classically automatic, hypothesis, a continuous homomorphism from R onto a compact metric abelian group is periodic, but that the existence of the minimum value of the period is not derivable (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The bornology of cofinally complete subsets

Cofinal completeness of a metric space, which is a property between completeness and compactness, can be characterized in terms of a measure of local compactness functional [7]. Using this functional, we introduce and then study the variational notion of cofinally complete subset of a metric space.     

Constructive Order Theory

We introduce the notion of constructive suprema and of constructively directed sets. The Axiom of Choice turns out to be equivalent to the postulate that every supremum is constructive, but also to the hypothesis that every directed set admits a function assigning to each finite subset an upper bound. The Axiom of Multiple Choice (which is known to be weaker than the full Axiom of Choice in set theory without foundation) implies a simple set‐theoretical induction principle (SIP), stating that any system of sets that is closed under unions of well‐ordered subsystems and contains all finite subsets of a given set must also contain that set itself. This is not provable without choice principles but equivalent to the statement that the existence of joins for constructively directed subsets of a poset follows from the existence of joins for nonempty well‐ordered subsets. Moreover, we establish the equivalence of SIP with several other fundamental statements concerning inductivity, compactness, algebraic closure systems, and the exchange between chains and directed sets.     

Bounded linear mappings of finite rank

This paper comprises a constructive investigation of the relationship between compactness, finite rank and located kernel for a bounded linear mapping into a finite-dimensional normed space. The main result is that a bounded linear mapping of a normed space into a finite-dimensional normed space is constructively compact if and only if its kernel is located. Several examples are given which highlight the constructive distinction between a mapping into a finite-dimensional space and a mapping with finite-dimensional range.     

Between compactness and completeness

Call a sequence in a metric space cofinally Cauchy if for each positive ε there exists a cofinal (rather than residual) set of indices whose corresponding terms are ε-close. We give a number of new characterizations of metric spaces for which each cofinally Cauchy sequence has a cluster point. For example, a space has such a metric if and only each continuous function defined on it is uniformly locally bounded. A number of results exploit a measure of local compactness functional that we introduce. We conclude with a short proof of Romaguera's Theorem: a metrizable space admits such a metric if and only if its set of points having a compact neighborhood has compact complement.     

Sobolev spaces on an arbitrary metric measure space: Compactness of embeddings

We formulate a new definition of Sobolev function spaces on a domain of a metric space in which the doubling condition need not hold. The definition is equivalent to the classical definition in the case that the domain lies in a Euclidean space with the standard Lebesgue measure. The boundedness and compactness are examined of the embeddings of these Sobolev classes into L _q and C _α . We state and prove a compactness criterion for the family of functions L _p (U), where U is a subset of a metric space possibly not satisfying the doubling condition.     

下方图度量下的非紧模糊数空间

证明了非紧模糊数空间E^～在下方图度量下关于模糊数的序是可逼近的。本文给出的证明方法是构造性的，从而说明了模糊数值积分如M-积分和G-积分等是可计算的。最后给出了E^～中关于下方图度量的一些分析性质。     

An Extension of the Contraction Principle

The concept of quasi-continuity and the new concept of almost compactness for a function are the basis for the extension of the contraction principle in large deviations presented here. Important equivalences for quasi-continuity are proved in the case of metric spaces. The relation between the exponential tightness of a sequence of stochastic processes and the exponential tightness of its transform (via an almost compact function) is studied here in metric spaces. Counterexamples are given to the nonmetric case. Relations between almost compactness of a function and the goodness of a rate function are studied. Applications of the main theorem are given, including to an approximation of the stochastic integral.