Locatedness and overt sublocales

Locatedness is one of the fundamental notions in constructive mathematics. The existence of a positivity predicate on a locale, i.e. the locale being overt, or open, has proved to be fundamental in constructive locale theory. We show that the two notions are intimately connected.Bishop defines a metric space to be compact if it is complete and totally bounded. A subset of a totally bounded set is again totally bounded iff it is located. So a closed subset of a Bishop compact set is Bishop compact iff it is located. We translate this result to formal topology. ‘Bishop compact’ is translated as compact and overt. We propose a definition of locatedness on subspaces of a formal topology, and prove that a closed subspace of a compact regular formal space is located iff it is overt. Moreover, a Bishop-closed subset of a complete metric space is Bishop compact — that is, totally bounded and complete — iff its localic completion is compact overt.Finally, we show by elementary methods that the points of the Vietoris locale of a compact regular locale are precisely its compact overt sublocales.We work constructively, predicatively and avoid the use of the axiom of countable choice.     

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.     

Metric complements of overt closed sets

We show that the set of points of an overt closed subspace of a metric completion of a Bishop‐locally compact metric space is located. Consequently, if the subspace is, moreover, compact, then its collection of points is Bishop‐compact. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Compactness under constructive scrutiny

How are the various classically equivalent definitions of compactness for metric spaces constructively interrelated? This question is addressed with Bishop‐style constructive mathematics as the basic system – that is, the underlying logic is the intuitionistic one enriched with the principle of dependent choices. Besides surveying today's knowledge, the consequences and equivalents of several sequential notions of compactness are investigated. For instance, we establish the perhaps unexpected constructive implication that every sequentially compact separable metric space is totally bounded. As a by‐product, the fan theorem for detachable bars of the complete binary fan proves to be necessary for the unit interval possessing the Heine‐Borel property for coverings by countably many possibly empty open balls. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Wijsman topology on function spaces

LetC(X,Y) be the space of continuous functions from a metric space (X,d) to a metric space (Y, e).C(X, Y) can be thought as subset of the hyperspaceCL(X×Y) of closed and nonempty subsets ofX×Y by identifying each element ofC(X,Y) with its graph. We considerC(X,Y) with the topology inherited from the Wijsman topology induced onCL(X×Y) by the box metric ofd ande. We study the relationships between the Wijsman topology and the compact-open topology onC(X,Y) and also conditions under which the Wijsman topology coincide with the Fell topology. Sufficient conditions under which the compactopen topology onC(X,Y) is weaker than the Wijsman topology are given (IfY is totally bounded, then for every metric spaceX the compactopen topology onC(X,Y) is weaker than the Wijsman topology and the same is true forX locally connected andY rim-totally bounded). We prove that a metric spaceX is boundedly compact iff the Wijsman topology onC(X, ℝ) is weaker than the compact-open topology. We show that ifX is a σ-compact complete metric space andY a compact metric space, then the Wijsman topology onC(X,Y) is Polish.     

Super weak compactness and uniform Eberlein compacta

We prove that a topological space is uniform Eberlein compact iff it is homeomorphic to a super weakly compact subset C of a Banach space such that the closed convex hull coC of C is super weakly compact. We also show that a Banach space X is super weakly compactly generated iff the dual unit ball B_X* of X^* in its weak star topology is affinely homeomorphic to a super weakly compactly convex subset of a Banach space.     

Wijsman topology on function spaces

LetC(X,Y) be the space of continuous functions from a metric space (X,d) to a metric space (Y, e).C(X, Y) can be thought as subset of the hyperspaceCL(X×Y) of closed and nonempty subsets ofX×Y by identifying each element ofC(X,Y) with its graph. We considerC(X,Y) with the topology inherited from the Wijsman topology induced onCL(X×Y) by the box metric ofd ande. We study the relationships between the Wijsman topology and the compact-open topology onC(X,Y) and also conditions under which the Wijsman topology coincide with the Fell topology. Sufficient conditions under which the compactopen topology onC(X,Y) is weaker than the Wijsman topology are given (IfY is totally bounded, then for every metric spaceX the compactopen topology onC(X,Y) is weaker than the Wijsman topology and the same is true forX locally connected andY rim-totally bounded). We prove that a metric spaceX is boundedly compact iff the Wijsman topology onC(X, ?) is weaker than the compact-open topology. We show that ifX is a σ-compact complete metric space andY a compact metric space, then the Wijsman topology onC(X,Y) is Polish.     

Finitary sequence spaces

This paper studies the metric structure of the space H_r of absolutely summable sequences of real numbers with at most r nonzero terms. H_r is complete, and is located and nowhere dense in the space of all absolutely summable sequences. Totally bounded and compact subspaces of H_r are characterized, and large classes of located, totally bounded, compact, and locally compact subspaces are constructed. The methods used are constructive in the strict sense. MSC: 03F65, 54E50.     

On local non‐compactness in recursive mathematics

A metric space is said to be locally non‐compact if every neighborhood contains a sequence that is eventually bounded away from every element of the space, hence contains no accumulation point. We show within recursive mathematics that a nonvoid complete metric space is locally non‐compact iff it is without isolated points. The result has an interesting consequence in computable analysis: If a complete metric space has a computable witness that it is without isolated points, then every neighborhood contains a computable sequence that is eventually computably bounded away from every computable element of the space. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Completions of partial metrics into value lattices

In this paper we investigate some notions of completion of partial metric spaces, including the bicompletion, the Smyth completion, and a new “spherical completion”. Given an auxiliary relation, we show that it arises from a totally bounded partial metric space, and the spherical completion of such a space is its round ideal completion. We also give an example of a totally bounded partial metric space whose bicompletion and Smyth completion are not continuous posets. Finally, we present an example of a totally bounded partial metric giving rise to the Scott and lower topologies of a continuous poset, but whose spherical completion is not a continuous poset.     

Constructive irrational space

The Fréchet combination allows the construction of a complete metric on the set of irrational numbers. The constructive study of the resulting spaceM was begun by Errett Bishop. This paper studies the structure ofM in some detail. The constructive approach requires a strong form of the concept of irrational number and particular attention to the distinctions between the various notions of points exterior to a set. The main results are the characterization and construction of all compact and locally compact subspaces ofM.     

A methodology for the constructive approximation of nonlinear operators defined on noncompact sets

We consider the constructive approximation of a non-linear operator that is known on a bounded but not necessarily compact set. Our main result can be regarded as an extension of the classical Stone-Weierstrass Theorem and also shows that the approximation is stable to small

disturbances.

This problem arises in the modelling of real dynamical systems where an input-output mapping is known only on some bounded subset of the input space. In such cases it is desirable to construct a model of the real system with a complete input-output map that preserves, in some approximate sense, the known mapping. The model is normally constructed from an algebra of elementary continuous functions.

We will assume that the input space is a separable Hilbert space. To solve the problem we introduce a special weak topology and show that uniform continuity of the given operator in the weak topology provides an alternative compactness condition that is sufficient to justify the desired approximation.     

An Analog of Titchmarsh’s Theorem for the Fourier–Walsh Transform

The space clos(X) of all nonempty closed subsets of an unbounded metric space X is considered. The space clos(X) is endowed with a metric in which a sequence of closed sets converges if and only if the distances from these sets to a fixed point θ are bounded and, for any r, the sequence of the unions of the given sets with the exterior balls of radius r centered at θ converges in the Hausdorff metric. The metric on clos(X) thus defined is not equivalent to the Hausdorff metric, whatever the initial metric space X. Conditions for a set to be closed, totally bounded, or compact in clos(X) are obtained; criteria for the bounded compactness and separability of clos(X) are given. The space of continuous maps from a compact space to clos(X) is considered; conditions for a set to be totally bounded in this space are found.     

Formal continuity implies uniform continuity near compact images on metric spaces

The localic completion of a metric space induces a canonical notion of continuous map between metric spaces. It is shown that these maps are continuous in the sense of Bishop constructive mathematics, i.e., uniformly continuous near every compact image.     

Resolution of the uniform lower bound problem in constructive analysis

In a previous paper we constructed a full and faithful functor ?? from the category of locally compact metric spaces to the category of formal topologies (representations of locales). Here we show that for a real‐valued continuous function f, ??(f) factors through the localic positive reals if, and only if, f has a uniform positive lower bound on each ball in the locally compact space. We work within the framework of Bishop constructive mathematics, where the latter notion is strictly stronger than point‐wise positivity. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

局部有限超拓扑的局部紧性

本文讨论了赋予局部有限拓扑的非空闲子集超空间的局部紧性.主要结果是:X正则,则其闭子集超空间局部紧当且仅当X可表示成一个紧空间与一个离散空间的拓扑和.     

A note on closed images of locally compact metric spaces

Summary A decomposition theorem about closed images of locally compact metric spaces is discussed. It is shown that a space is a closed image of a locally compact metric space if and only if it is a regular Fréchet space with a point-countable k-network, and each of its closed first-countable subset is locally compact.     

关于凸子集和（绝对）收缩核

讨论了线性度量空间中凸子集在什么情况下为该空间的收缩核，以及在什么情况下为绝对收缩核。     

Fixed point, approximate fixed point and Kantorovich-Rubinstein maximum principle in convex metric spaces

We obtain necessary conditions for the existence of fixed point and approximate fixed point of nonexpansive and asymptotically nonexpansive maps defined on a closed bounded convex subset of a uniformly convex complete metric space and study the structure of the set of fixed points. We construct Mann type iterative sequences in convex metric space and study its convergence. As a consequence of fixed point results, we prove best approximation results. We also prove Kantorovich-Rubinstein maximum principle in convex metric spaces.