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)     

On metric spaces with the Haver property which are Menger spaces

A metric space (X,d) has the Haver property if for each sequence ?₁,?₂,… of positive numbers there exist disjoint open collections V₁,V₂,… of open subsets of X, with diameters of members of Vi less than ?i and covering X, and the Menger property is a classical covering counterpart to σ-compactness. We show that, under Martin's Axiom MA, the metric square (X,d)×(X,d) of a separable metric space with the Haver property can fail this property, even if X² is a Menger space, and that there is a separable normed linear Menger space M such that (M,d) has the Haver property for every translation invariant metric d generating the topology of M, but not for every metric generating the topology. These results answer some questions by L. Babinkostova [L. Babinkostova, When does the Haver property imply selective screenability? Topology Appl. 154 (2007) 1971-1979; L. Babinkostova, Selective screenability in topological groups, Topology Appl. 156 (1) (2008) 2-9].     

On Zippin's Embedding Theorem of Banach spaces into Banach spaces with bases

We present a new proof of Zippin's Embedding Theorem, that every separable reflexive Banach space embeds into one with shrinking and boundedly complete basis, and every Banach space with a separable dual embeds into one with a shrinking basis. This new proof leads to improved versions of other embedding results.     

Final compactness and separability in regular symmetrizable spaces

In 1966 A. V. Arkhangel'skii posed the following question: Is it true that every regular finally compact symmetrizable space is separable? S. I. Nedev soon showed that a regular finally compact symmetrizable space is hereditarily finally compact. Consequently any counterexample to Arkhangel'skii's conjecture must be an L-space. Applying the technique of iterated forcing we prove that in the axiom systemZFC for set theory it is consistent to assume the existence of a regular (hereditarily) finally compact symmetrizable space X that is nonseparable. Thus it is impossible to prove using the axiom systemZFC that every regular finally compact symmetrizable space is separable. The space X has additional properties as well: it has a basis consisting of open/closed sets (i.e., it is zero-dimensional in the sense ofind, it can be mapped continuously and one-to-one onto a separable metric space, it is α-left and has cardinality ω₁. Bibliography: 25 titles.     

Reverse mathematics of separably closed sets

This paper contains a corrected proof that the statement “every non-empty closed subset of a compact complete separable metric space is separably closed” implies the arithmetical comprehension axiom of reverse mathematics.     

On some results of analysis for fuzzy metric spaces

A necessary and sufficient condition for a fuzzy metric space to be complete is given. We prove that a subspace of a separable fuzzy metric space is separable and every separable fuzzy metric space is second countable. Uniform limit theorem is generalized to fuzzy metric spaces.     

Lebesgue numbers and Atsuji spaces in subsystems of second-order arithmetic

We study Lebesgue and Atsuji spaces within subsystems of second order arithmetic. The former spaces are those such that every open covering has a Lebesgue number, while the latter are those such that every continuous function defined on them is uniformly continuous. The main results we obtain are the following: the statement “every compact space is Lebesgue” is equivalent to ; the statements “every perfect Lebesgue space is compact” and “every perfect Atsuji space is compact” are equivalent to ; the statement “every Lebesgue space is Atsuji” is provable in ; the statement “every Atsuji space is Lebesgue” is provable in . We also prove that the statement “the distance from a closed set is a continuous function” is equivalent to . Received: February 2, 1996     

Stable maps of Polish spaces

We define the notions of stable and transquotient maps and study the relation between these classes of maps. The class of stable maps contains all closed and open maps and their compositions. The transquotient maps preserve the property of being a Polish space, and every stable map between separable metric spaces is transquotient.

In particular, a composition of closed and open maps (the intermediary spaces may not be metric) preserves the property of being a Polish space. This generalizes the results of Sierpinski and Vainstein for open and closed maps.

     

Sigma-compact,nowhere locally compact metric spaces can be densely imbedded in Hilbert space

It is shown that if H is a connected, locally contractible, separable, topologically complete metric space with the property that mappings of separable metric spaces into H are approximable by imbeddings (in particular, if H is Hilbert space), then every sigma-compact, nowhere locally compact metric space can be densely imbedded in H.     

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.     

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.     

Embeddings of countable closed sets and reverse mathematics

Summary If there is a homeomorphic embedding of one set into another, the sets are said to be topologically comparable. Friedman and Hirst have shown that the topological comparability of countable closed subsets of the reals is equivalent to the subsystem of second order arithmetic denoted byATR ₀. Here, this result is extended to countable closed locally compact subsets of arbitrary complete separable metric spaces. The extension uses an analogue of the one point compactification of .     

A model of intuitionistic analysis in which ‐definable discrete sets are subcountable

There is a model, for a system of intuitionistic analysis including Brouwer's principle for numbers and Kripke's schema, in which ‐definable discrete sets of choice sequences are subcountable.     

Set-theoretic constructions of nonshrinking open covers

A family {M_α|α?A} is a shrinking of a cover {O_α|α?A} of a topological space if {M_α|α?A} also covers and M_α?O_α for all α?A.?⁺⁺ implies that there is a normal space such that every increasing open cover of it has a clopen shrinking but there is an open cover having no closed shrinking.? implies that there is a P-space (i.e. a space having a normal product with every metric space), which has an increasing open cover having no closed shrinking. This space is used in [17] to show that any space which has a normal product with every P-space is metrizable.     

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.     

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)     

Constructive notions of equicontinuity

In the informal setting of Bishop-style constructive reverse mathematics we discuss the connection between the antithesis of Specker’s theorem, Ishihara’s principle BD-N, and various types of equicontinuity. In particular, we prove that the implication from pointwise equicontinuity to uniform sequential equicontinuity is equivalent to the antithesis of Specker’s theorem; and that, for a family of functions on a separable metric space, the implication from uniform sequential equicontinuity to uniform equicontinuity is equivalent to BD-N.      

Strongly e-movable convergence and spaces of ANR's

Let X be a metric space and let ANR(X) denote the hyperspace of all compact ANR's in X. This paper introduces a notion of a strongly e-movable convergence for sequences in ANR(X) and proves several characterizations of strongly e-movable convergence. For a (complete) separable metric space X we show that ANR(X) with the topology induced by strongly e-movable convergence can be metrized as a (complete) separable metric space. Moreover, if X is a finite-dimensional compactum, then strongly e-movable convergence induces on ANR(X) the same topology as that induced by Borsuk's homotopy metric.For a separable Q-manifold M, ANR(M) is locally arcwise connected and A, B ? ANR(M) can be joined by an arc in ANR(M) iff there is a simple homotopy equivalence ?: A → B homotopic to the inclusion of A into M.     

A nilregular element property

An element a of a commutative ring R is nilregular if and only if x is nilpotent whenever ax is nilpotent. More generally, an ideal I of R is nilregular if and only if x is nilpotent whenever ax is nilpotent for all a ∈ I . We give a direct proof that if R is Noetherian, then every nilregular ideal contains a nilregular element. In constructive mathematics, this proof can then be seen as an algorithm to produce nilregular elements of nilregular ideals whenever R is coherent, Noetherian, and discrete. As an application, we give a constructive proof of the Eisenbud-Evans-Storch theorem that every algebraic set in n-dimensional affine space is the intersection of n hypersurfaces.Received: 6 September 2004     

A metric characterization of the irrationals via a group operation

Let S be a separable metric space with a compatible metric d that satisfies: For each point x ? S and each nonnegative real number r there exists a unique point y ? S such that d(x,y) = r.In this paper spaces that meet the above criterion are investigated. It is shown that, under the assumption of completeness, this metric property characterizes the space of irrationals.