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In this paper, we present a way of constructing a topology on a vertex set of a graph using monophonic eccentric neighborhoods of the graph G. In this type of construction, we characterize those graphs that induced the indiscrete topology, the discrete topology, and a particular point topology.

Quite recently, Wu and Liu (2020) [11] raised an open problem when they discussed the relationships between generalized rough sets and topologies. Said precisely, each binary relation generates a topology through the lower rough approximation operator, then for two binary relations on the same set, is there a sufficient and necessary condition such that the union of generated topologies by two binary relations, is identical with the generated topology by the join of two binary relations. In this note, we give a positive answer to this problem and provide a union-join condition. Furthermore, note that the main reason that the expected identity hold not consists in the fact that the union of two topologies is not a topology, in general. Then replacing the union of two topologies with the supremum of them, we give a sufficient and necessary condition (supremum-join condition) such that the supremum of generated topologies by two binary relations, is identical with the generated topology by the join of two binary relations. We verify that the supremum-join condition is much simpler than the union-join condition. At last, we give a characterization on discrete topology generated by a binary relation using the sober separation.

In the recent paper R. A. Muneshwar and K. L. Bondar, introduced a graph topological structure, called open subset inclusion graph of a topological space j(t ) on a finite set X. In the present pap...

Motivated by the work of Frazier; and Gabardo and Nashed, we study $$P^{th}$$ -stage nonuniform discrete wavelet frames ( $$P^{th}$$ -stage NUDW frames, in short) for $$\ell ^2(\Lambda )$$ , a nonuniform discrete space. In nonuniform discrete wavelet frames, the translation set is not necessary a group but a spectrum which is based on the theory of spectral pairs. We characterize first-stage nonuniform discrete Bessel sequences and wavelet frames in nonuniform discrete spaces. Duality and stability of first-stage NUDW frames are also discussed. Finally, by using first-stage NUDW frames, we provide a suitable way to construct the $$P^{th}$$ -stage NUDW frames. We illustrate our construction with the help of a concrete example.

The concept of topology has great significance in the study of spatial objects and their associated problems encountered in Physics. In topology, a discrete space is an example of a topological space in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. In this paper, we introduce the concept of Discrete Metric Graphs which is derived from the concept of Discrete Metric Space in Topology. Let G(V,E) be an undirected graph, then the Discrete Metric Graph of G denoted by D(G) is the graph whose any two vertices u and v are connected if deg(u)≠deg(v) in G. The properties of the Discrete Metric Graphs of some standard graphs such as path, regular graph, cycle, complete graph, complete bipartite graph, star, wheel, etc… have been studied in this paper.

In order to extract geometric information from images, suitable operators must be constructed. After a discussion of convexity and geodesic distance, the important notion of metric dilation is introduced, followed by that of distance transforms. Sections are then devoted to geodesic and conditional operators, granulometries, connectivity and skeletons. A final section considers discrete metric spaces.

We define pseudocompleteness in the category of locales in a conservative way; so that a space is pseudocomplete in the sense of Oxtoby [24] if and only if the locale it determines is pseudocomplete. We show that a pseudocomplete locale whose G δ -sublocales are complemented (for instance if it is scattered) is a Baire locale in the sense of Isbell [20] . Our main theorem is that products of pseudocomplete locales are pseudocomplete. Whereas every discrete space is pseudocomplete, and Boolean locales generalize discrete spaces, we demonstrate that not every Boolean locale is pseudocomplete. In [27] Pichardo-Mendoza asks whether pseudocompleteness (in topological spaces) is an invariant of closed irreducible maps. We answer this in the affirmative.

We obtain an independence result connected to the classic Moore–Mrowka problem. A property known to be intermediate between sequential and countable tightness in the class of compact spaces is the notion of a space being C-closed. A space is C-closed if every countably compact subset is closed. We prove it is consistent to have a compact C-closed space that is not sequential. Our example also answers a question of Arhangelskii by producing a compactification of the countable discrete space which is not itself sequential and yet it has a Frechet–Urysohn remainder. Ismail and Nyikos showed that compact C-closed spaces are sequential if $${2^{\mathfrak{t}} > 2^\omega}$$ . We prove that compact C-closed spaces are sequential also holds in the standard Cohen model.

The Hewitt–Marczewski–Pondiczery theorem states that if X = is the Tychonoff product, where d ( X α ) ≤ τ ≥ ω for all α ∈ A and | A | ≤ 2 τ , then d ( X ) ≤ τ . We prove that in the product of discrete spaces D α = ω there is a dense countable subset Q = ∪ { Q k : k ∈ ω } , the union of disjoint finite sets Q k , satisfying the following conditions: – if F ⊆ Q is such that | F ∩ Q k | ≤ 1 for all k ∈ ω , then F is a discrete closed in set; – if C ⊆ ω is infinite and for F ⊆ Q there is m 0 ∈ ω such that | Q k ∩ F | ≤ m 0 for all k ∈ ω , then ∪ { Q k : k ∈ C } ∖ F is dense in ; – Q contains no convergent in non-trivial sequences

Compact Hausdorff topological MV-algebras and Stone MV-algebras are completely characterized. We obtain that compact Hausdorff topological MV-algebras are product (both topological and algebraic) of copies $[0,1]$ with standard topology and finite Lukasiewicz chains with discrete topology. Going one step further we also prove that Stone MV-algebras are product (both topological and algebraic) of finite Lukasiewicz chains with discrete topology. We also prove that an MV-algebra is strongly complete (isomorphic to its profinite completion) if and only if it is profinite and its maximal ideals of finite ranks are principal., 15 pages

Fuzzy extractors (Dodis et al., Eurocrypt 2004) convert repeated noisy readings of a high-entropy source into the same uniformly distributed key. The functionality of a fuzzy extractor outputs the key when provided with a value close to the original reading of the source. A necessary condition for security, called fuzzy min-entropy, is that the probability of every ball of values of the noisy source is small.Many noisy sources are best modeled using continuous metric spaces. To build continuous-source fuzzy extractors, prior work assumes that the system designer has a good model of the distribution (Verbitskiy et al., IEEE TIFS 2010). However, it is impossible to build an accurate model of a high entropy distribution just by sampling from the distribution.Model inaccuracy may be a serious problem. We demonstrate a family of continuous distributions ${\mathcal{W}}$ that is impossible to secure. No fuzzy extractor designed for ${\mathcal{W}}$ extracts a meaningful key from an average element of ${\mathcal{W}}$. This impossibility result is despite the fact that each element $W \in {\mathcal{W}}$ has high fuzzy min-entropy. We show a qualitatively stronger negative result for secure sketches, which are used to construct most fuzzy extractors.Our results are for the Euclidean metric and are information-theoretic in nature. To the best of our knowledge all continuous-source fuzzy extractors argue information-theoretic security.Fuller, Reyzin, and Smith showed comparable negative results for a discrete metric space equipped with the Hamming metric (Asiacrypt 2016). Continuous Euclidean space necessitates new techniques.

The paper is devoted to the geometry of transportation cost spaces and their generalizations introduced by Melleray et al. (Fundam Math 199(2):177–194, 2008). Transportation cost spaces are also known as Arens–Eells, Lipschitz-free, or Wasserstein 1 spaces. In this work, the existence of metric spaces with the following properties is proved: (1) uniformly discrete metric spaces such that transportation cost spaces on them do not contain isometric copies of $$\ell _1$$, this result answers a question raised by Cuth and Johanis (Proc Am Math Soc 145(8):3409–3421, 2017); (2) locally finite metric spaces which admit isometric embeddings only into Banach spaces containing isometric copies of $$\ell _1$$; (3) metric spaces for which the double-point norm is not a norm. In addition, it is proved that the double-point norm spaces corresponding to trees are close to $$\ell _\infty ^d$$ of the corresponding dimension, and that for all finite metric spaces M, except a very special class, the infimum of all seminorms for which the embedding of M into the corresponding seminormed space is isometric, is not a seminorm.

A finite sequence a → = 〈 a i 〉 i = 1 m in Z ∖ { 0 } is compressed provided a i ≠ a i + 1 for i m . It is known that if a → = 〈 a i 〉 i = 1 m and b → = 〈 b i 〉 i = 1 k are compressed sequences in Z ∖ { 0 } , then there exist idempotents p and q in β Q d ∖ { 0 } such that a 1 p + a 2 p + … + a m p = b 1 q + b 2 q + … + b k q if and only if b → is a rational multiple of a → . In fact, if b → is not a rational multiple of a → , then there is a partition of Q ∖ { 0 } into two cells, neither of which is a member of a 1 p + a 2 p + … + a m p and a member of b 1 q + b 2 q + … + b k q for any idempotents p and q in β Q d ∖ { 0 } . (Here β Q d is the Stone–Cech compactification of the set of rational numbers with the discrete topology.) In this paper we extend these results to direct sums of Q . As a corollary, we show that if b → is not a rational multiple of a → and G is any torsion free commutative group, then there do not exist idempotents p and q in β G d ∖ { 0 } such that a 1 p + a 2 p + … + a m p = b 1 q + b 2 q + … + b k q . We also show that for direct sums of finitely many copies of Q we can separate the corresponding Milliken–Taylor systems, with a similar but weaker result for the direct sum of countably many copies of Q .

The Standard Simplex Conjecture and the Plurality is Stablest Conjecture are two conjectures stating that certain partitions are optimal with respect to Gaussian and discrete noise stability respectively. These two conjectures are natural generalizations of the Gaussian noise stability result by Borell (1985) and the Majority is Stablest Theorem (2004). Here we show that the standard simplex is not the most stable partition in Gaussian space and that Plurality is not the most stable low influence partition in discrete space for every number of parts k ≥ 3, for every value ρ ≠ 0 of the noise and for every prescribed measure for the different parts as long as they are not all equal to 1/k. Our results do not contradict the original statements of the Plurality is Stablest and Standard Simplex Conjectures in their original statements concerning partitions to sets of equal measure. However, they indicate that if these conjectures are true, their veracity and their proofs will crucially rely on assuming that the sets are of equal measures, in stark contrast to Borell’s result, the Majority is Stablest Theorem and many other results in isoperimetric theory. Given our results it is natural to ask for (conjectured) partitions achieving the optimum noise stability.

In this paper, we propose a definition of approximation property which is called the metric invariant translation approximation property for a countable discrete metric space. Moreover, we use the techniques of Ozawa’s to prove that a fine hyperbolic graph has the metric invariant translation approximation property.

A set A of natural numbers is finitely embeddable in another such set B if every finite subset of A has a rightward translate that is a subset of B. This notion of finite embeddability arose in combinatorial number theory, but in this paper we study it in its own right. We also study a related notion of finite embeddability of ultrafilters on the natural numbers. Among other results, we obtain connections between finite embeddability and the algebraic and topological structure of the Stone-Cech compactification of the discrete space of natural numbers. We also obtain connections with nonstandard models of arithmetic., to appear in Bulletin of the Polish Academy of Sciences, Math Series

The theory of limit operators was developed by Rabinovich, Roch and Silbermann to study the Fredholmness of band-dominated operators on $\ell^p(\mathbb{Z}^N)$ for $p \in \{0\} \cup [1,\infty]$, and recently generalised to discrete metric spaces with property A by \v{S}pakula and Willett for $p \in (1,\infty)$. In this paper, we study the remained extreme cases of $p \in\{0,1,\infty\}$ (in the metric setting) to fill the gaps., Comment: 26 pages

We now move on to general continuous-time Markov processes on countable spaces, i.e., we consider processes \((X_t)_{t>0}\) for which the time variable may assume any value in \([0,\infty )\) (or some bounded interval), and each X(t) takes its values in some countable space \(\mathcal{S}\). While the concept of a Markov transition semigroup introduces itself as a generalization of the one-step transition matrix of a discrete-time process, the task of extracting information on the dynamics of the process now becomes significantly harder. The Kolmogorov equations will be derived by essentially analytical methods, which (as before in the case of the Poisson process) rely on dicretizing the process, which means using skeletons. In order to keep the presentation elementary, some of the results we give are not optimal, and whenever necessary, references are given for sharper results.

We consider the Stone space of the Boolean algebra, constructed by M.G. Bell (see Example 2.1 in [2] ), which is the compactification, BN , of the countable discrete space N . Here we consider convergent sequences in B N ∖ N . We prove that if a point x ∈ B N ∖ N is the limit of some sequence { s n : n ∈ ω } from N , or a point x ∈ [ A ] ∖ A , where A ⊆ N is a strict anti-chain in N , then there is a sequence in B N ∖ N , that converges to x . We also prove that if A is a countable discrete set of u -points in B N ∖ N and x ∈ [ A ] ∖ A , then x is not the limit of any sequence of points of B N ∖ N .

Given a stream of rectangles over a discrete space, we consider the problem of computing the total number of distinct points covered by the rectangles. This is the discrete version of the two-dimensional Klee's measure problem for streaming inputs. Given 0 < ? , ? < 1 , we provide ( ? , ? ) -approximations for bounded side length rectangles and for bounded aspect ratio rectangles. For the case of arbitrary rectangles, we provide an O ( log ? U ) -approximation, where U is the total number of discrete points in the two-dimensional space. The time to process each rectangle and the total required space are polylogarithmic in U. The time to answer a query for the total area is constant. We construct efficient transformation techniques that project rectangle areas to one-dimensional ranges and then use a streaming algorithm for the one-dimensional Klee's measure problem to obtain these approximations. The projections are deterministic, and to our knowledge, these are the first approaches of this kind that provide efficiency and accuracy trade-offs in the streaming model.

We study the class H of all topological abelian groups ( G , τ ) with the property that the only locally compact group topology on G strictly finer than τ is the discrete topology. Hewitt showed that the reals R and the circle group T belong to H and Rickert and Rajagopalan characterized the non-discrete locally compact groups in the class H as those that contain either T , R , or the compact group J p of p-adic integers as an open subgroup. We describe the entire class H , providing many examples of non-locally compact groups with this property. This allows us to obtain as a by-product the results of Hewitt, Rajagopalan and Rickert and give some applications to characterized subgroups.

We prove that, for every n , the topological space ω n ω (where ω n has the discrete topology) can be partitioned into ℵ n copies of the Baire space. Using this fact, we then prove two new theorems about completely ultrametrizable spaces. We say that Y is a condensation of X if there is a continuous bijection f : X → Y . First, it is proved that ω ω is a condensation of ω n ω if and only if ω ω can be partitioned into ℵ n Borel sets, and some consistency results are given regarding such partitions. It is also proved that it is consistent with ZFC that, for any n ω , c = ω n and there are exactly n + 3 similarity types of perfect completely ultrametrizable spaces of size c . These results answer two questions of the first author from [1] .

This paper deals with the problem of fitting a discrete polynomial curve to 2D noisy data. We use a discrete polynomial curve model achieving connectivity in the discrete space. We formulate the fitting as the problem to find parameters of this model maximizing the number of inliers i.e., data points contained in the discrete polynomial curve. We propose a method guaranteeing inclusion-wise maximality of its obtained inlier set.

There exist two known concepts of ultrafilter extensions of first-order models, both in a certain sense canonical. One of them [1] comes from modal logic and universal algebra, and in fact goes back to [2]. Another one [3, 4] comes from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups [5] as its main precursor. By a classical fact, the space of ultrafilters over a discrete space is its largest compactification. The main result of [3, 4], which confirms a canonicity of this extension, generalizes this fact to discrete spaces endowed with a first-order structure. An analogous result for the former type of ultrafilter extensions was obtained in [6].

Let X be an unbounded metric space, B ( x , r ) = { y ∈ X : d ( x , y ) ≤ r } for all x ∈ X and r ≥ 0 . We endow X with the discrete topology and identify the Stone–Cech compactification βX of X with the set of all ultrafilters on X . Our aim is to reveal some features of algebra in βX similar to the algebra in the Stone–Cech compactification of a discrete semigroup [6] . We denote X # = { p ∈ β X : each P ∈ p is unbounded in X } and, for p , q ∈ X # , write p ∥ q if and only if there is r ≥ 0 such that B ( Q , r ) ∈ p for each Q ∈ q , where B ( Q , r ) = ⋃ x ∈ Q B ( x , r ) . A subset S ⊆ X # is called invariant if p ∈ S and q ∥ p imply q ∈ S . We characterize the minimal closed invariant subsets of X , the closure of the set K ( X # ) = ⋃ { M : M is a minimal closed invariant subset of X # } , and find the number of all minimal closed invariant subsets of X # . For a subset Y ⊆ X and p ∈ X # , we denote △ p ( Y ) = Y # ∩ { q ∈ X # : p ∥ q } and say that a subset S ⊆ X # is an ultracompanion of Y if S = △ p ( Y ) for some p ∈ X # . We characterize large, thick, prethick, small, thin and asymptotically scattered spaces in terms of their ultracompanions.

For simulation of objects of discrete topology, supercomputers are often used. This can be explained by the combinatorial nature of the problems examined, especially in the study of the dynamics of rearrangements of such structures. In this paper, we consider three methods of coding associated with the parallelization of computations: (1) coding of cubic complexes; (2) coding for the calculation of transition probabilities in Markov chains in the reconstructions of triangulations of Euclidean spaces; (3) coding of topological situations in the transformations of complexes, i.e., discrete analogs of homotopic transformation. The base chosen for the geometric and topological objects possesses “minimal” mathematical resources (integer points and primitive vectors in Euclidean spaces).

We prove that if X is a quasiregular, countably compact space with a pseudobase consisting of closed G δ -sets, then every G δ -dense subspace of X is pseudocomplete in the sense of Todd. In particular, every weakly pseudocompact space is pseudocomplete in the sense of Todd. Some sufficient conditions are found that guarantee that a weakly pseudocompact space is pseudocomplete in the sense of Oxtoby. It is shown that every weakly pseudocompact space without isolated points has cardinality at least continuum. An example is given of a weakly pseudocompact space with one non-isolated point that contains the one-point lindelofication of an uncountable discrete space. We apply this example to show that weak pseudocompactness is not preserved by the relation of M -equivalence.

We characterize the compactness properties of the product of λ copies of the space ω with the discrete topology, dealing in particular with the case λ singular, using regular and uniform ultrafilters, infinitary languages and nonstandard elements. We also deal with products of uncountable regular cardinals with the order topology.

Let μ be a point on a countable discrete space . Under the assumption that μ is quasi-invariant with respect to any finitary permutation of , we describe a general scheme for constructing an equilibrium Kawasaki dynamics for which μ is a symmetrizing (and hence invariant) measure. We also exhibit a two-parameter family of point process μ possessing the needed quasi-invariance property. Each process of this family is determinantal, and its correlation kernel is the kernel of a projection in l2 ( ). Bibliography: 17 titles.

A natural duality is obtained for each finitely generated variety B n ( n p -algebras. The duality for B n is based on a schizophrenic object: P −1 in B n is the algebra 2 n ⊕ 1 which generates the variety and P −1 is a topological relational structure carrying the discrete topology and a set of algebraic relations. The relations are (i) the graphs of a (3-element) generating set for the endomorphism monoid of P −1 and (ii) a set of subalgebras of P 2 −2 in one-to-one correspondence with partitions of the integer n . Each of the latter class of relations, regarded as a digraph, is ‘nearly’ the union of two isomorphic trees. The duality is obtained by the piggyback method of Davey and Werner (which has previously yielded a duality in case n ≤ 2), combined with use of the restriction to finite p -algebras of the duality for bounded distributive lattices, which enables the relations suggested by the general theory to be concretely described.

International audience; Random variables with Mittag-Leffler distribution can take values either in the set of non-negative integers or in the positive real line. There can be of two different types, one (type-1) heavy-tailed with index α ∈ (0, 1), the other (type-2) possessing all its moments. We investigate various stochastic processes where they play a key role, among which: the discrete space/time Neveu branching process, the discrete-space continuous-time Neveu branching process, the continuous space/time Neveu branching process (CSBP) and renewal processes with rare events. Its relation to (discrete or continuous) self-decomposability and branching processes with immigration is emphasized. Special attention will be paid to the Neveu CSBP for its connection with the Bolthausen-Sznitman coalescent. In this context, and following a recent work of Möhle [49], a type-2 Mittag-Leffler process turns out to be the Siegmund dual to Neveu's CSBP block-counting process arising in sampling from P D e −t , 0. Further combinatorial developments of this model are investigated. 1. Sibuya random variables (rvs) and related branching processes We first investigate a class of integral-valued rvs that will show important for our general purpose. 1.1. Sibuya rvs and related ones. We start with their definition and main properties. • One parameter Sibuya(α) rv. Let X α ≥ 1 be an integer-valued random variable with support N = {1, 2, ....} defined as follows: X α = inf (l ≥ 1 : B α (l) = 1) , where (B α (l)) l≥1 is a sequence of independent Bernoulli rvs obeying P (B α (l) = 1) = α/l where α ∈ (0, 1). It is thus the first epoch of a success in a Bernoulli trial when the probability of success is inversely proportional to the number of the trial. X α is called a Sibuya(α) rv. Then P (X α = k) = (−1)

Given the pairwise distances for a set of unknown points in a known metric space, the distance geometry problem DGP is to compute the point coordinates in conformation with the distance constraints. It is a well-known problem in the Euclidean space, has several variations, finds many applications, and so has been attempted by different researchers from time to time. However, to the best of our knowledge, it is not yet fully addressed to its merit, especially in the discrete space. Hence, in this paper we introduce a novel variant of DGP where the pairwise distance between every two unknown points is given a tolerance zone with the objective of finding the solution as a collection of integer points. The solution is based on characterization of different types of annulus intersection, their equivalence, and cardinality bounds of integer points. Necessary implementation details and useful heuristics make it attractive for practical applications in the discrete space.

The properties of splittability and divisibility introduced by A.V. Arhangel’skii and cardinal functions defined on their basis are considered. The growth of split weight and divisibility degree is examined under taking topological products.

Let D be a countable discrete space and let p∈D. For any primitive shift σ:D→D∖{p} there are 2c σ-invariant maximal almost disjoint families on D. This implies that there are 2c pairwise non-homeomorphic Ψ⁎ spaces admitting primitive shifts. Under a=c there exists a maximal almost disjoint family F on a countable discrete space D such that the spaces Ψ(F) and Ψ⁎(F) admit no primitive shift.

A space X has the Collins–Roscoe property if we can assign, to each x ∈ X , a family G ( x ) of subsets of X in such a way that for every set A ⊂ X , the family ⋃ { G ( a ) : a ∈ A } contains an external network of A ¯ . Every space with the Collins–Roscoe property is monotonically monolithic. We show that for any uncountable discrete space D , the space C p ( β D ) does not have the Collins–Roscoe property; since C p ( β D ) is monotonically monolithic, this proves that monotone monolithity does not imply the Collins–Roscoe property and provides an answer to two questions of Gruenhage. However, if X is a Lindelof Σ -space with n w ( X ) ⩽ ω 1 then C p ( X ) has the Collins–Roscoe property; this implies that C p ( X ) is metalindelof and constitutes a generalization of an analogous theorem of Dow, Junnila and Pelant proved for a compact space X . We also establish that if X and C p ( X ) are Lindelof Σ -spaces, then the iterated function space C p , n ( X ) has the Collins–Roscoe property for every n ∈ ω .

Given a dynamical system ( X , f ) with X a compact metric space and a free ultrafilter p on N , we define f p ( x ) = p - lim n → ∞ f n ( x ) for all x ∈ X . It was proved by A. Blass (1993) that x ∈ X is recurrent iff there is p ∈ N ⁎ = β ( N ) ∖ N such that f p ( x ) = x . This suggests to consider those points x ∈ X for which f p ( x ) = x for some p ∈ N ⁎ , which are called p-recurrent. We shall give an example of a recurrent point which is not p-recurrent for several p ∈ N ⁎ . Also, A. Blass proved that two points x , y ∈ X are proximal iff there is p ∈ N ⁎ such that f p ( x ) = f p ( y ) (in this case, we say that x and y are p-proximal). We study the properties of the p-proximal points of the following continuous self maps of the Cantor set: For an arbitrary function f : N → N , we define σ f : { 0 , 1 } N → { 0 , 1 } N by σ f ( x ) ( k ) = x ( f ( k ) ) for every k ∈ N and for every x ∈ { 0 , 1 } N (the shift map on { 0 , 1 } N is obtained by the function k ↦ k + 1 ). Let E ( X ) denote the Ellis semigroup of the dynamical system ( X , f ) . We prove that if f : N → N is a function with at least one infinite orbit, then E ( { 0 , 1 } N , σ f ) is homeomorphic to β ( N ) . Two functions g , h : N → N are defined so that E ( { 0 , 1 } N , σ g ) is homeomorphic to the Cantor set, and E ( { 0 , 1 } N , σ h ) is the one-point compactification of N with the discrete topology.

We continue our investigations on the lattice ( q ( X ) , ⊆ ) of quasi-uniformities on a set X . Improving on earlier results, we show that the Pervin quasi-uniformity (resp. the well-monotone quasi-uniformity) of an infinite topological T 1 -space X does not have a complement in ( q ( X ) , ⊆ ) . We also establish that a hereditarily precompact quasi-uniformity inducing the discrete topology on an infinite set X does not have a complement in ( q ( X ) , ⊆ ) .

Given the discrete space of natural numbers, we characterize the elements of polynomials evaluated on the points of βℕ. We establish these results by proving the characterization in a far more general setting. Let S be a discrete set which is a semigroup under two operations ⋅ and +. Let g(z1,z2,…,zk) be any polynomial and p1,p2,…,pk be elements of βS. We provide a sufficient condition that a set A⊆S is a member of g(p1,p2,…,pk) and use it to characterize the members of g(p1,p2,…,pk) if each pi is an idempotent in (βS,+).

In this paper, we study the group H u ( R ) of uniform homeomorphisms having the uniform topology together with two subgroups H ∞ ( R ) = { h ∈ H u ( R ) ; lim | x | → ∞ ( h ( x ) − x ) = 0 } and H c ( R ) , the group of compact support homeomorphisms. We show that the group H u ( R ) is homeomorphic to l ∞ × 2 ℵ 0 and that the triple ( H u ( R ) 0 , H ∞ ( R ) , H c ( R ) ) is homeomorphic to ( l ∞ × l 2 × l 2 , { 0 } × l 2 × l 2 , { 0 } × l 2 × l 2 f ) , where 2 ℵ 0 denotes a discrete space of the cardinality of the continuum, while H u ( R ) 0 is the identity component of H u ( R ) and l 2 f is the linear hull of the standard orthonormal basis of the separable Hilbert space l 2 . We will also discuss the relations among H u ( R ) 0 and some function spaces containing it.

Let X be a countable discrete metric space and let XX denote the family of all functions on X. In this article, we consider the problem of finding the least cardinality of a subset A of XX such that every element of XX is a finite composition of elements of A and Lipschitz functions on X. It follows from a classical theorem of Sierpiński that such an A either has size at most 2 or is uncountable.We show that if X contains a Cauchy sequence or a sufficiently separated, in some sense, subspace, then |A|≤1. On the other hand, we give several results relating |A| to the cardinal d; defined as the minimum cardinality of a dominating family for NN. In particular, we give a condition on the metric of X under which |A|≥d holds and a further condition that implies |A|≤d. Examples satisfying both of these conditions include all subsets of Nk and the sequence of partial sums of the harmonic series with the usual euclidean metric.To conclude, we show that if X is any countable discrete subset of the real numbers R with the usual euclidean metric, then |A|=1 or almost always, in the sense of Baire category, |A|=d.

For a space X, 2 X denotes the collection of all non-empty closed sets of X with the Vietoris topology, and K ( X ) denotes the collection of all non-empty compact sets of X with the subspace topology of 2 X . The following are known: • 2 ω is not normal, where ω denotes the discrete space of countably infinite cardinality. • For every non-zero ordinal γ with the usual order topology, K ( γ ) is normal iff cf γ = γ whenever cf γ is uncountable. In this paper, we will prove: (1) 2 ω is strongly zero-dimensional. (2) K ( γ ) is strongly zero-dimensional, for every non-zero ordinal γ. In (2), we use the technique of elementary submodels.

Two models for analyzing the process of extraction (and impregnation) from a porous material are considered. Cells, i.e., discrete space descriptions, are used in the models. The first model also assumes a discrete time description. In the second model, time varies continuously. It is shown that these two models and a conventional (diffusion) model form a hierarchical chain of models. Relationships for calculating the quantity of the target component in a porous solid are derived.

The paper aims to establish topological links between perception of objects (as it is defined in the framework of near sets) and classification of these objects (as it is defined in the framework of rough sets). In the near set approach, the discovery of near sets (i.e. sets containing objects with similar descriptions) starts with the selection of probe functions which provide a basis for describing and discerning objects. On the other hand, in the rough set approach, the classification of objects is based on object attributes which are collected into information systems (or data tables). As is well-known, an information system can be represented as a topological space (U, τ E). If we pass froman approximation space (U,E) to the quotient space U/E, where points represent indiscernible objects of U, then U/E will be endowed with the discrete topology induced (via the canonical projection) by τ E. The main objective of this paper is to show how probe functions can provide new topologies on the quotient set U/E and, in consequence, new (perceptual) topologies on U.

Preserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. In this context, we present an introductory study of the new notion of simple set which extends the classical notion of simple point. Similarly to simple points, simple sets have the property that the homotopy type of the object in which they lie is not changed when such sets are removed. Simple sets are studied in the framework of cubical complexes which enables, in particular, to model the topology in Z^n. The main contributions of this article are: a justification of the study of simple sets (motivated by the limitations of simple points); a definition of simple sets and of a subfamily of them called minimal simple sets; the presentation of general properties of (minimal) simple sets in n-D spaces, and of more specific properties related to ''small dimensions'' (these properties being devoted to be further involved in studies of simple sets in 2,3 and 4-D spaces).

Let $${\beta(\mathbb{N})}$$ denote the Stone–Cech compactification of the set $${\mathbb{N}}$$ of natural numbers (with the discrete topology), and let $${\mathbb{N}^\ast}$$ denote the remainder $${\beta(\mathbb{N})-\mathbb{N}}$$ . We show that, interpreting modal diamond as the closure in a topological space, the modal logic of $${\mathbb{N}^\ast}$$ is S4 and that the modal logic of $${\beta(\mathbb{N})}$$ is S4.1.2.

The notion of an ideal family of weighted subspaces of a discrete metric space X with bounded geometry is introduced. It is shown that, if X has Yu’s property A, the ideal structure of the Roe algebra of X with coefficients in \( \mathcal{B} \)(H) is completely characterized by the ideal families of weighted subspaces of X, where \( \mathcal{B} \)(H) denotes the C*-algebra of bounded linear operators on a separable Hilbert space H.

International audience; In this paper we study the relationship between the Euclidean and the discrete space. We study discrete operations based on Euclidean functions: the discrete smooth scaling and the discrete-continuous rotation. Conversely, we study Euclidean oper- ations based on discrete functions: the discrete based simplification, the Euclidean- discrete union and the Euclidean-discrete co-refinement. These operations operate partly in the discrete and partly in the continuous space. Especially for the discrete smooth scaling operation, we provide error bounds when different such operations are chained.

AnR-moduleMis called a multiplication module if for each submoduleNofM,N=IMfor some idealIofR. As defined for a commutative ringR, anR-moduleMis said to be semiprimitive if the intersection of maximal submodules ofMis zero. The maximal spectra of a semiprimitive multiplication moduleMare studied. The isolated points of Max(M) are characterized algebraically. The relationships among the maximal spectra ofM, Soc(M) and Ass(M) are studied. It is shown that Soc(M) is exactly the set of all elements ofMwhich belongs to every maximal submodule ofMexcept for a finite number. If Max(M) is infinite, Max(M) is a one-point compactification of a discrete space if and only ifMis Gelfand and for some maximal submoduleK, Soc(M) is the intersection of all prime submodules ofMcontained inK. WhenMis a semiprimitive Gelfand module, we prove that every intersection of essential submodules ofMis an essential submodule if and only if Max(M) is an almost discrete space. The set of uniform submodules ofMand the set of minimal submodules ofMcoincide. Ann(Soc(M))Mis a summand submodule ofMif and only if Max(M) is the union of two disjoint open subspacesAandN, whereAis almost discrete andNis dense in itself. In particular, Ann(Soc(M)) = Ann(M) if and only if Max(M) is almost discrete.

Let (X, ~) be a combinatorial graph the vertex set X of which is a discrete metric space. We suppose that a discrete group G acts freely on (X, ~) and that the fundamental domain with respect to the action of G contains only a finite set of points. A graph with these properties is called periodic with respect to the group G. We examine the Fredholm property and the essential spectrum of band-dominated operators acting on the spaces l p (X) or c_0(X), where (X, ~) is a periodic graph. Our approach is based on the thorough use of band-dominated operators. It generalizes the necessary and sufficient results obtained in [39] in the special case $$X = G = {\mathbb{Z}}^{n}$$ and in [42] in case X = G is a general finitely generated discrete group.