Weak regularity and consecutive topologizations and regularizations of pretopologies

Abstract L. Foged proved that a weakly regular topology on a countable set is regular. In terms of convergence theory, this means that the topological reflection Tξ of a regular pretopology ξ on a countable set is regular. It is proved that this still holds if ξ is a regular σ -compact pretopology. On the other hand, it is proved that for each n ω there is a (regular) pretopology ρ (on a set of cardinality c ) such that ( RT ) k ρ > ( RT ) n ρ for each k n and ( RT ) n ρ is a Hausdorff compact topology, where R is the reflector to regular pretopologies. It is also shown that there exists a regular pretopology of Hausdorff RT -order ⩾ ω 0 . Moreover, all these pretopologies have the property…

Introduction to General Duality Theory for Multi-Objective Optimization

This is intended as a comprehensive introduction to the duality theory for vector optimization recently developed by C. Malivert and the present author [3]. It refers to arbitrarily given classes of mappings (dual elements) and extends the general duality theory proposed for scalar optimization by E. Balder, S. Kurcyusz and the present author [1] and P. Lindberg.

The forgotten mathematical legacy of Peano

International audience; The formulations that Peano gave to many mathematical notions at the end of the 19th century were so perfect and modern that they have become standard today. A formal language of logic that he created, enabled him to perceive mathematics with great precision and depth. He described mathematics axiomatically basing the reasoning exclusively on logical and set-theoretical primitive terms and properties, which was revolutionary at that time. Yet, numerous Peano’s contributions remain either unremembered or underestimated.

Precise bounds for the sequential order of products of some Fréchet topologies

Abstract The sequential order of a topological space is the least ordinal for which the corresponding iteration of the sequential closure is idempotent. Lower estimates for the sequential order of the product of two regular Frechet topologies and upper estimates for the sequential order of the product of two subtransverse topologies are given in terms of their fascicularity and sagittality. It is shown that for every countable ordinal α, there exists a Lasnev topology such that the sequential order of its square is equal to α.

Erratum to “Irregularity” [Topology Appl. 154 (8) (2007) 1565–1580]

[2, Proposition 4.4] states that each regular pretopology is topologically regular. Professor F. Mynard (Georgia Southern University) advised the authors that he was not convinced by the proof of that proposition, which enabled us to realize the proposition is wrong, as the example below shows. Recall that (e.g., [2]) a pretopology ξ on a set X is called regular if Vξ (x)⊂ adh ξ Vξ (x) (respectively, topologically regular if Vξ (x)⊂ cl ξ Vξ (x)) for every x ∈ X . As a consequence, in the sequel of [2], regular should be read topologically regular in a few instances, in particular in [2, Theorem 4.6]. [2, Proposition 4.4] is also quoted in [3], where it is used in some reformulations of clas…

Completeness number of families of subsets of convergence spaces

International audience; Compactoid and compact families generalize both convergent filters and compact sets. This concept turned out to be useful in various quests, like Scott topologies, triquotient maps and extensions of the Choquet active boundary theorem.The completeness number of a family in a convergence space is the least cardinality of collections of covers for which the family becomes complete. 0-completeness amounts to compactness, finite completeness to relative local compactness and countable completeness to Čech completeness. Countably conditional countable completeness amounts to pseudocompleteness of Oxtoby. Conversely, each completeness class of families can be represented a…

Convergence foundations of topology

International audience

Cascades and multifilters

Abstract Cascades (trees every element of which is a filter on the set of its successors), and multifilters, maps from cascades, are introduced. Multisequences constitute a special case of multifilters. Applications to convergence and to topology are indicated.

Irregularity

AbstractRegular and irregular pretopologies are studied. In particular, for every ordinal there exists a topology such that the series of its partial (pretopological) regularizations has length of that ordinal. Regularity and topologicity of special pretopologies on some trees can be characterized in terms of sets of intervals of natural numbers, which reduces studied problems to combinatorics.

Group topologies coarser than the Isbell topology

Abstract The Isbell, compact-open and point-open topologies on the set C ( X , R ) of continuous real-valued maps can be represented as the dual topologies with respect to some collections α ( X ) of compact families of open subsets of a topological space X . Those α ( X ) for which addition is jointly continuous at the zero function in C α ( X , R ) are characterized, and sufficient conditions for translations to be continuous are found. As a result, collections α ( X ) for which C α ( X , R ) is a topological vector space are defined canonically. The Isbell topology coincides with this vector space topology if and only if X is infraconsonant. Examples based on measure theoretic methods, t…

Some old and new results on lower semicontinuity of minimal points

Polarities and Generalized Extremal Convolutions

International audience;

When a convergence of filters is measure-theoretic

Abstract Convergence almost everywhere cannot be induced by a topology, and if measure is finite, it coincides with almost uniform convergence and is finer than convergence in measure, which is induced by a metrizable topology. Measures are assumed to be finite. It is proved that convergence in measure is the Urysohn modification of convergence almost everywhere, which is pseudotopological. Extensions of these convergences from sequences to arbitrary filters are discussed, and a concept of measure-theoretic convergence is introduced. A natural extension of convergence almost everywhere is neither measure-theoretic, nor finer than a natural extension of convergence in measure. A straightforw…

Convergence-theoretic mechanisms behind product theorems

Abstract Commutation of the topologizer with products, quotientness of product maps, preservation of some properties by products, topologicity of continuous convergence, continuity of complete lattices are facets of the same quest. A new method of multifilters is used to establish (in terms of core-contour-compactness) sufficient and necessary conditions for these properties in the framework of general convergences. The relativized Antoine reflector plays here an important role. Several classical results (of Whitehead, Michael, Boehme, Cohen, Day and Kelly, Hofmann and Lawson, Schwarz and Weck, Kent and Richardson, and others) are extended or refined.

Convergence-theoretic characterizations of compactness

AbstractFundamental variants of compactness are characterized in terms of concretely reflective convergence subcategories: topologies, pretopologies, paratopologies, hypotopologies and pseudotopologies. Hyperquotient maps (perfect, quasi-perfect, adherent and closed) and quotient maps (quotient, hereditarily quotient, countably biquotient, biquotient, and almost open) are characterized in terms of various degrees of compactness of their fiber relations, and of sundry relaxations of inverse continuity.

Multiple facets of inverse continuity

International audience; Inversion of various inclusions that characterize continuity in topological spaces results in numerous variants of quotient and perfect maps. In the framework of convergences, the said inclusions are no longer equivalent, and each of them characterizes continuity in a different concretely reflective subcategory of convergences. On the other hand, it turns out that the mentioned variants of quotient and perfect maps are quotient and perfect maps with respect to these subcategories. This perspective enables use of convergence-theoretic tools in quests related to quotient and perfect maps, considerably simplifying the traditional approach. Similar techniques would be un…

General duality in vector optimization

Vector minimization of a relation F valued in an ordered vector space under a constraint A consists in finding x 0 ∊ A w,0 ∊ Fx$0 such that w,0 is minimal in FA. To a family of vector minimization problemsminimize , one associates a Lagrange relation where ξ belongs to an arbitrary class Ξ of mappings, the main purpose being to recover solutions of the original problem from the vector minimization of the Lagrange relation for an appropriate ξ. This ξ turns out to be a solution of a dual vector maximization problem. Characterizations of exact and approximate duality in terms of vector (generalized with respect to Ξ) convexity and subdifferentiability are given. They extend the theory existin…