Continuity of posets via Scott topology and sobrification

In this paper, posets which may not be dcpos are considered. The concept of embedded bases for posets is introduced. Characterizations of continuity of posets in terms of embedded bases and Scott topology are given. The main results are:

(1)

A poset is continuous iff it is an embedded basis for a dcpo up to an isomorphism;

(2)

A poset is continuous iff its Scott topology is completely distributive;

(3)

A topological T₀ space is a continuous poset equipped with the Scott topology in the specialization order iff its topology is completely distributive and coarser than or equal to the Scott topology;

(4)

A topological T₁ space is a discrete space iff its topology is completely distributive.

These results generalize the relevant results obtained by J.D. Lawson for dcpos.     

Constructive heuristics for the uncapacitated continuous location-allocation problem

The multisource location-allocation problem in continuous space is investigated. Two constructive heuristic techniques are proposed to solve this problem. Both methods are based on designing suitable schemes for the generation of the initial solutions. The first considers the furthest distance rule and is enhanced by schemes borrowed from tabu search such as constructing the forbidden regions and freeing strategy. The second considers the discrete solutions found when solving the p-median problem. Some results on existing test problems are presented.     

Über mittlere Rückkehrzeit unter einer ma\treuen Strömung

Summary Let T be a one-parameter semigroup of measure preserving transformations of a probability space. The theorem of Kac on mean recurrence time of the points x in a measurable subset E under a discrete semigroup is carried over to the case of a flow T with continuous time parameter t0. Recurrence time is defined as the infimum of these parameter values t>0 for which the orbit of x has returned to E after having temporarily left the set E. The results are first formulated for a probability space without any topological structure; they are then applied to the case of a continuous flow in a compact metric space.     

Topological entropy for set valued maps

Any continuous map T on a compact metric space X induces in a natural way a continuous map on the space K(X) of all non-empty compact subsets of X. Let T be a homeomorphism on the interval or on the circle. It is proved that the topological entropy of the induced set valued map is zero or infinity. Moreover, the topological entropy of is zero, where C(X) denotes the space of all non-empty compact and connected subsets of X. For general continuous maps on compact metric spaces these results are not valid.     

Continuous reducibility and dimension of metric spaces

If (X, d) is a Polish metric space of dimension 0, then by Wadge’s lemma, no more than two Borel subsets of X are incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space (X, d) of positive dimension, there are uncountably many Borel subsets of (X, d) that are pairwise incomparable with respect to continuous reducibility. In general, the reducibility that is given by the collection of continuous functions on a topological space \((X,\tau )\) is called the Wadge quasi-order for \((X,\tau )\). As an application of the main result, we show that this quasi-order, restricted to the Borel subsets of a Polish space \((X,\tau )\), is a well-quasiorder if and only if \((X,\tau )\) has dimension 0. Moreover, we give further examples of applications of the construction of graph colorings that is used in the proofs.     

Fixed point set of semigroups of non-expansive mappings and amenability

In this paper, we study the fixed point set of a strongly continuous non-expansive semigroup of a semi-topological semigroup S for which CB(S) is n-extremely left amenable. Also, we study the fixed point set of a strongly continuous semigroup of mappings when S is a semigroup which is a sub-semigroup of a locally convex topological vector space with addition. Some applications to harmonic analysis are also provided.     

On pointwise approximation of arbitrary functions by countable families of continuous functions

The following problem is considered. Given a real-valued function f defined on a topological space X, when can one find a countable familyf _n :n∈ω of continuous real-valued functions on X that approximates f on finite subsets of X? That is, for any finite set F?X and every real number ε>0 one can choosen∈ω such that ∥f(x)?f_n(x)∥<ε for everyx∈F. It will be shown that the problem has a positive solution if and only if X splits. A space X is said to split if, for any A?X, there exists a continuous mapf _A:X→R ^ω such that A=f _A ^?1 (A). Splitting spaces will be studied systematically.     

Between perception and intuition: Learning about infinity

Based on an empirical study, we explore children’s primary and secondary perceptions on infinity. When discussing infinity, children seem to highlight three categories of primary perceptions: processional, topological, and spiritual. Based on their processional perception, children see the set of natural numbers as being infinite and endow Q with a discrete structure by making transfers from N to Q. In a continuous context, children are more likely to mobilize a topological perception. Evidence for a secondary perception of N arises from students’ propensities to develop infinite sequences of natural numbers, and from their ability to prove that N is infinite. Children’s perceptions on infinity change along the school years. In general, the perceptual dominance moves from sequential (processional) to topological across development. However, we found that around 11-13 years old, processional and topological perceptions interfere with each other, while before and after this age they seem to coexist and collaborate, one or the other being specifically activated by the nature of different tasks.     

Full characterizations of minimax inequality,fixed point theorem,saddle point theorem,and KKM principle in arbitrary topological spaces

This paper provides necessary and sufficient conditions for the existence of solutions for some important problems from optimization and non-linear analysis by replacing two typical conditions—continuity and quasiconcavity with a unique condition, weakening topological vector spaces to arbitrary topological spaces that may be discrete, continuum, non-compact or non-convex. We establish a single condition, \(\gamma \)-recursive transfer lower semicontinuity, which fully characterizes the existence of \(\gamma \)-equilibrium of minimax inequality without imposing any restrictions on topological space. The result is then used to provide full characterizations of fixed point theorem, saddle point theorem, and KKM principle.     

Continuity of the Spectrum of a Field of Self-Adjoint Operators

Given a family of self-adjoint operators \({(A_t)_{t \in T}}\) indexed by a parameter t in some topological space T, necessary and sufficient conditions are given for the spectrum \({\sigma(A_t)}\) to be Vietoris continuous with respect to t. Equivalently the boundaries and the gap edges are continuous in t. If (T, d) is a complete metric space with metric d, these conditions are extended to guarantee Hölder continuity of the spectral boundaries and of the spectral gap edges. As a corollary, an upper bound is provided for the size of closing gaps.     

Topologies on abelian lattice ordered groups induced by a positive filter and completeness

We consider topologies on an abelian lattice ordered group that are determined by the absolute value and a positive filter. We show that the topological completions of these objects are also determined by the absolute value and a positive filter. We investigate the connection between the topological completion of such objects and the Dedekind–MacNeille completion of the underlying lattice ordered group. We consider the preservation of completeness for such topologies with respect to homomorphisms of lattice ordered groups. Finally, we show that topologies defined in terms of absolute value and a positive filter on the space C(X) of all real-valued continuous functions defined on a completely regular topological space X are always complete.     

Discretization orders for distance geometry problems

Given a weighted, undirected simple graph G = (V, E, d) (where \({d:E\to\mathbb{R}_+}\)), the distance geometry problem (DGP) is to determine an embedding \({x:V\to\mathbb{R}^K}\) such that \({\forall \{i,j\} \in E\;\|x_i-x_j\|=d_{ij}}\) . Although, in general, the DGP is solved using continuous methods, under certain conditions the search is reduced to a discrete set of points. We give one such condition as a particular order on V. We formalize the decision problem of determining whether such an order exists for a given graph and show that this problem is NP-complete in general and polynomial for fixed dimension K. We present results of computational experiments on a set of protein backbones whose natural atomic order does not satisfy the order requirements and compare our approach with some available continuous space searches.     

Injective spaces via adjunction

The work of the present author and his coauthors over the past years gives evidence that it may be useful to regard each topological space as a kind of enriched category, by interpreting the convergence relation x→x between ultrafilters and points of a topological space X as arrows in X. Naturally, this point of view opens the door to the use of concepts and ideas from enriched Category Theory for the investigation of topological spaces. Topological theories introduced by the author provide a convenient general setting for appropriately transferring these concepts and ideas to the world of topological spaces and some other geometric objects such as approach spaces. Using tools like adjunction and the Yoneda lemma, we show that the cocomplete spaces are precisely the injective spaces, and they are algebras for a suitable monad on . This way we obtain enriched versions of known results about injective topological spaces and continuous lattices.     

An arithmetical view to first-order logic

A value space is a topological algebra B equipped with a non-empty family of continuous quantifiers :B^∗→B. We will describe first-order logic on the basis of B. Operations of B are used as connectives and its relations are used to define statements. We prove under some normality conditions on the value space that any theory in the new setting can be represented by a classical first-order theory.     

On integration with respect to a DT-measure

DT-measures generalize the topological measures of Aarnes. We continue to study an integral of a real-valued continuous function on compact Hausdorff space with respect to a DT-measure and investigate the following: multiplicativity of this integral, \(w*\)-topology on the family of DT-measures and density of various subfamilies of this space, and also integral as a set function.     

Average-value tverberg partitions via finite fourier analysis

The long-standing topological Tverberg conjecture claimed, for any continuous map from the boundary of an N(q, d):= (q-1)(d+1)-simplex to d-dimensional Euclidian space, the existence of q pairwise disjoint subfaces whose images have non-empty q-fold intersection. The affine cases, true for all q, constitute Tverberg’s famous 1966 generalization of the classical Radon’s Theorem. Although established for all prime powers in 1987 by Özaydin, counterexamples to the conjecture, relying on 2014 work of Mabillard and Wagner, were first shown to exist for all non-prime-powers in 2015 by Frick. Starting with a reformulation of the topological Tverberg conjecture in terms of harmonic analysis on finite groups, we show that despite the failure of the conjecture, continuous maps below the tight dimension N(q, d) are nonetheless guaranteed q pairwise disjoint subfaces–including when q is not a prime power–which satisfy a variety of “average value” coincidences, the latter obtained as the vanishing of prescribed Fourier transforms.     

Resolutions of topological linear spaces and continuity of linear maps

The main result of the paper is the following: If an F-space X is covered by a family of sets such that Eα⊂Eβ whenever α?β, and f is a linear map from X to a topological linear space Y which is continuous on each of the sets Eα, then f is continuous. This provides a very strong negative answer to a problem posed recently by J. Ka?kol and M. López Pellicer. A number of consequences of this result are given, some of which are quite curious. Also, inspired by a related question asked by J. Ka?kol, it is shown that if a linear map is continuous on each member of a sequence of compact sets, then it is also continuous on every compact convex set contained in the linear span of the sequence. The construction applied to prove this is then used to interpret a natural linear topology associated with the sequence as the inductive limit topology in the sense of Ph. Turpin, and thus derive its basic properties.     

On the Closure of the Product and Sum of Linear Relations

In this paper we give two topological conditions on the graphs of two linear relations E and F on a Hilbert space ${\mathcal H}$ , one of which is sufficient to ensure the closure of the product EF, the other being sufficient to ensure the closure of the sum E?+?F. Note that these conditions are not necessary and in particular do not include the cases when E and F are continuous everywhere defined operators. Other related results are established.