Sequentially Compact Research Articles

A Pedagogical History of Compactness

This paper traces the history of compactness from the original motivating questions through the development of the definition to a characterization of compactness in terms of nets and filters. The goal of the article is to clarify the central concepts of open-cover and sequential compactness, including details that a standard textbook treatment tends to leave out.

COUNTABILITY AND APPROACH THEORY

In approach theory, we can provide arbitrary products of <TEX>${\infty}p$</TEX>-metric spaces with a natural structure, whereas, classically only if we rely on a countable product and the question arises, then, whether properties which are derived from countability properties in metric spaces, such as sequential and countable compactness, can also do away with countability. The classical results which simplify the study of compactness in pseudometric spaces, which proves that all three of the main kinds of compactness are identical, suggest a further study of the category <TEX>$pMET^{\infty}$</TEX>.

Topological spaces compact with respect to a set of filters

Abstract If is a family of filters over some set I, a topological space X is sequencewise -compact if for every I-indexed sequence of elements of X there is such that the sequence has an F-limit point. Countable compactness, sequential compactness, initial κ-compactness, [λ; µ]-compactness, the Menger and Rothberger properties can all be expressed in terms of sequencewise -compactness for appropriate choices of . We show that sequencewise -compactness is preserved under taking products if and only if there is a filter such that sequencewise -compactness is equivalent to F-compactness. If this is the case, and there exists a sequencewise -compact T 1 topological space with more than one point, then F is necessarily an ultrafilter. The particular case of sequential compactness is analyzed in detail.

Generalized sequential normal compactness in Asplund spaces

Sequential normal compactness conditions are important properties in infinite-dimensional variational analysis and its applications. Following the recent study of the generalized sequential normal compactness (GSNC), this paper This paper reveals further applications of GSNC to the generalized differentiation theory in Asplund spaces, as well as the calculus of GSNC itself.

A New Characterization of Compact Sets in Fuzzy Number Spaces

We give a new characterization of compact subsets of the fuzzy number space equipped with the level convergence topology. Based on this, it is shown that compactness is equivalent to sequential compactness on the fuzzy number space endowed with the level convergence topology. Our results imply that some previous compactness criteria are wrong. A counterexample also is given to validate this judgment.

Generalized sequential normal compactness in Banach spaces

Normal compactness conditions are important properties of sets, set-valued mappings in variational analysis, which are generalized versions of the classical Lipschitzian property and are essential for the calculus of generalized differentiation theory. In this paper we propose the notion called the generalized sequential normal compactness, and establish its basic properties and calculus in general Banach spaces.

Sequentially injective and complete acts over a semigroup

In this paper using the notion of a sequentially dense monomorphism we consider sequential injectivity (s-injectivity) for acts over a semigroup S. We show that s-injectivity, s-absolutely retract, and sequential compactness are equivalent.

Finiteness Theorems for Perfect Numbers and Their Kin

Since ancient times, a natural number has been called perfect if it equals the sum of its proper divisors; e.g., 6 = 1 + 2 + 3 is a perfect number. In 1913, Dickson showed that for each fixed k, there are only finitely many odd perfect numbers with at most k distinct prime factors. We show how this result, and many like it, follow from embedding the natural numbers in the supernatural numbers and imposing an appropriate topology on the latter; the notion of sequential compactness plays a starring role.

Reclassifying the antithesis of Specker’s theorem

It is shown that a principle, which can be seen as a constructivised version of sequential compactness, is equivalent to a form of Brouwer's fan theorem. The complexity of the latter depends on the geometry of the spaces involved in the former.

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.

A UNIFORM QUANTITATIVE FORM OF SEQUENTIAL WEAK COMPACTNESS AND BAILLON'S NONLINEAR ERGODIC THEOREM

We apply proof-theoretic techniques of "proof mining" to obtain an effective uniform rate of metastability in the sense of Tao for Baillon's famous nonlinear ergodic theorem in Hilbert space. In fact, we analyze a proof due to Brézis and Browder of Baillon's theorem relative to the use of weak sequential compactness. Using previous results due to the author we show the existence of a bar recursive functional Ω* (using only lowest type bar recursion B0, 1) providing a uniform quantitative version of weak compactness. Primitive recursively in this functional (and hence in T0+ B0, 1) we then construct an explicit bound φ on for the metastable version of Baillon's theorem. From the type level of φ and another result of the author it follows that φ is primitive recursive in the extended sense of Gödel's T. In a subsequent paper also Ω* will be explicitly constructed leading to the refined complexity estimate φ ∈ T4.

Subdifferential necessary conditions in set-valued optimization problems with equilibrium constraints

This article concerns new subdifferential necessary conditions for local optimal solutions to an important class of general set-valued optimization problems with abstract equilibrium constraints, where the optimality notion is understood in the sense of the generalized order from Definition 5.53 in the book ‘Variational Analysis and Generalized Differentiation II: Applications’ by B. Mordukhovich. This notion is induced by the concept of set extremality and covers all the conventional notions of optimality in vector optimization. Our method is mainly based on advanced tools of variational analysis and generalized differentiation. Our results are formulated in terms of the subdifferential constructions for set-valued mappings. We also provide several important relationships between subdifferentials and coderivatives of set-valued mappings including formulae, sequential normal compactness property, and Lipschitz-like behaviour which are new in vector optimization and even in scalar optimization.

Operators on the space of bounded strongly measurable functions

Let L ( X , Y ) stand for the space of all bounded linear operators between real Banach spaces ( X , ‖ ⋅ ‖ X ) and ( Y , ‖ ⋅ ‖ Y ) , and let Σ be a σ-algebra of subsets of a non-empty set Ω. Let L ∞ ( Σ , X ) denote the Banach space of all bounded strongly Σ-measurable functions f : Ω → X equipped with the supremum norm ‖ ⋅ ‖ . A bounded linear operator T from L ∞ ( Σ , X ) to a Banach space Y is said to be σ-smooth if ‖ T ( f n ) ‖ Y → 0 whenever ‖ f n ( ω ) ‖ X → 0 for all ω ∈ Ω and sup n ‖ f n ‖ < ∞ . It is shown that if an operator measure m : Σ → L ( X , Y ) is variationally semi-regular (i.e., m ˜ ( A n ) → 0 as A n ↓ ∅ , where m ˜ ( A ) stands for the semivariation of m on A ∈ Σ ), then the corresponding integration operator T m : L ∞ ( Σ , X ) → Y is σ-smooth. Conversely, it is proved that every σ-smooth operator T : L ∞ ( Σ , X ) → Y admits an integral representation with respect to its representing operator measure. We prove a Banach–Steinhaus type theorem for σ-smooth operators from L ∞ ( Σ , X ) to Y. In particular, we study the topological properties of the space L ∞ ( Σ , X ) c ⁎ of all σ-smooth functionals on L ∞ ( Σ , X ) . We prove a form of a generalized Nikodým convergence theorem and characterize relative σ ( L ∞ ( Σ , X ) c ⁎ , L ∞ ( Σ , X ) ) -sequential compactness in L ∞ ( Σ , X ) c ⁎ . We derive a Grothendieck type theorem for L ∞ ( Σ , X ) c ⁎ . The relationships between different classes of linear operators on L ∞ ( Σ , X ) are established.

The existence of optimal control on the basis of Weierstrass’s theorem

The problem of existence of an optimal control is solved on the basis of Weierstrass’s classical theorem if the set of admissible controls belongs to the class of piecewise continuous functions. In the process of describing admissible controls, the main assumption is that the number of switchings (points of discontinuity) is uniformly bounded and not just finite, as in the main problem of optimal control theory. On the one hand, this assumption does not restrict the spectrum of optimal control applications. On the other hand, it fits the Weierstrass’s theorem owing to the convenience in characterizing the sequential compactness. The formulation of Weierstrass’s theorem, which asserts the existence of continuous function extrema on sequentially compact sets, is customary, and its proof complies with the traditional scheme, whereas the concepts (convergent sequences and some others) are adapted to the peculiarity of optimal problems.

Erratum to: Ball-covering property of Banach spaces

Though several conclusions of [1] (i.e., Lemma 4.2, Theorem 4.3, Corollary 4.4, Theorem 4.5 and Theorem 4.7) are true in a renorming sense using a recent theorem either by V. Fonf and C. Zanco [5] or by L. Cheng, H. Shi and W. Zhang [3], the proofs of the mentioned results in [1] use w∗-separability of the unit ball BX∗ of a dual space X ∗ under the assumption that X∗ is w∗-separable and use w∗-sequential compactness of BX∗ with BX∗ being w∗-angelic. However, as pointed out by M. Fabian, w∗-separability of X∗ does not imply w∗-separability of BX∗ [6] (see, also [8]), and w ∗-separability of BX∗ does not entail that BX∗ is w∗-angelic in general. Recently, V. Kadets also informed me in an email that the closed unit ball of ∞ endowed with the equivalent norm, defined as the mean of the natural norm of ∞ and the quotient norm of /c0 , is such a counterexample (see his review of [4] in Zentralblatt Math.: Zbl 1152.46010). Recall a collection B of open (closed) balls in a Banach space X is said to be a ball-covering of X if every ball in B does not contain the origin and B covers the unit sphere SX of X . We say that X has the ball-covering property, if it admits a ball-covering of countably many balls. Some authors consider ball-coverings by open balls, the others — by closed ones. Since every open ball is a countable union of closed balls, and every closed ball off the origin is contained in an open ball off the origin with almost the same radius, we have the following simple property, which says that it does not matter much which of the two definitions one uses. ∗ Support by the Natural Science Foundation of China, grant 10771175. Received September 4, 2009 and in revised form February 19, 2010

Topological properties defined by nets

Nets are used to generalize a result of Murtinová and to define and study a class of properties related to sequential compactness.

Analyzing proofs based on weak sequential compactness

Analyzing proofs based on weak sequential compactness

FIXED POINT THEOREMS FOR GENERALIZED NONEXPANSIVE SET-VALUED MAPPINGS IN CONE METRIC SPACES

Abstract. In 2007, Huang and Zhang [1] introduced a cone metric spacewith a cone metric generalizing the usual metric space by replacing thereal numbers with Banach space ordered by the cone. They consideredsome xed point theorems for contractive mappings in cone metric spaces.Since then, the xed point theory for mappings in cone metric spaces hasbecome a subject of interest in [1-6] and references therein. In this paper,we consider some xed point theorems for generalized nonexpansive set-valued mappings under suitable conditions in sequentially compact conemetric spaces and complete cone metric spaces. 1. Introduction and PreliminariesIn 2007, Huang and Zhang [1] introduced a cone metric space with a conemetric generalizing the usual metric space by replacing the real numbers withBanach space ordered by the cone. They considered some xed point theoremsfor contractive mappings in cone metric spaces. Since then, the xed pointtheory for mappings in cone metric spaces has become a subject of interest in[1-6] and references therein. Especially, for single-valued mappings, Choudhuryand Metiya [6] considered some xed point theorems for weak contraction incone metric spaces in 2010. In 2011, Wardowski [2] introduced H-cone metricin the collection of subsets of a given cone metric space. And he consideredthe concept of set-valued contraction of Nadler-type and proved a xed pointtheorem for contractive set-valued mappings in H-cone metric spaces. Inspiredand encouraged by the previous works, in this paper we consider some xedpoint theorems for generalized nonexpansive set-valued mappings under suit-able conditions in sequentially compact cone metric spaces and complete conemetric spaces.Let Ebe a real Banach space and Pbe a subset of E.Pis called a cone if and only if

On quantitative versions of theorems due to F.E. Browder and R. Wittmann

This paper is another case study in the program of logically analyzing proofs to extract new (typically effective) information (‘proof mining’). We extract explicit uniform rates of metastability (in the sense of T. Tao) from two ineffective proofs of a classical theorem of F.E. Browder on the convergence of approximants to fixed points of nonexpansive mappings as well as from a proof of a theorem of R. Wittmann which can be viewed as a nonlinear extension of the mean ergodic theorem. The first rate is extracted from Browder's original proof that is based on an application of weak sequential compactness (in addition to a projection argument). Wittmann's proof follows a similar line of reasoning and we adapt our analysis of Browder's proof to get a quantitative version of Wittmann's theorem as well. In both cases one also obtains totally elementary proofs (even for the strengthened quantitative forms) of these theorems that neither use weak compactness nor the existence of projections anymore. In this way, the present article also discusses general features of extracting effective information from proofs based on weak compactness. We then extract another rate of metastability (of similar nature) from an alternative proof of Browder's theorem essentially due to Halpern that already avoids any use of weak compactness. The paper is concluded by general remarks concerning the logical analysis of proofs based on weak compactness as well as a quantitative form of the so-called demiclosedness principle. In a subsequent paper these results will be utilized in a quantitative analysis of Baillon's nonlinear ergodic theorem.

ON SPACES IN WHICH THE THREE MAIN KINDS OF COMPACTNESS ARE EQUIVALENT

In this paper, we introduce a new property (*) of a topological space and prove that if X satisfies one of the following conditions (1) and (2), then compactness, countable compactness and sequential compactness are equivalent in X; (1) Each countably compact subspace of X with (*) is a sequential or AP space. (2) X is a sequential or AP space with (*).