Search

We construct a counterexample that disproves the claim of Theorem 0.2 of the paper “Injective mappings and solvable vector fields of Euclidean spaces”, which appeared in Topology Appl. 136 (2004), 261–274. [ABSTRACT FROM AUTHOR]

The present paper studies digital topological rough set structures associated with Khalimsky (K -, for brevity) topology. We further explore their properties related to topological (closure and interior) operators. In addition, we find that the K -topological rough set structures have their own features different from certain theoretical properties of the H -rough set structure in [9] w.r.t. the topological operators. More precisely, whereas the H -rough set structure is a topological operator, K -topological concept approximations are proved not to be topological operators. Thus, this paper proposes an alternative approach to the K -topological concept approximations so that we finally have new types of topological operators. Furthermore, it appears that this new ones are smoothly matched with the earlier H -rough set structure from the viewpoint of topological operators. Therefore, they can facilitate the study of continuous-oriented or digital-oriented rough sets, which can be used in applied sciences without any limitations of the cardinalities of the universe and a target set. Besides, they can support certain decision rules with no restriction to anything continuous or discrete (or digital). In addition, we introduce the notion of K -definability of a target subset of a universal set and write an algorithm for estimating the roughness of the above new concept approximations. In this paper each set need not be finite and a covering is locally finite. [ABSTRACT FROM AUTHOR]

Let X be a continuum, K a nonempty closed subset of X , and let n be a positive integer. In this paper, we consider the hyperspaces 2 X and C n (X) , consisting of all nonempty closed subsets of X and of all nonempty closed subsets of X having at most n components, respectively. If H (X) ∈ { 2 X , C n (X) } , H (X ; K) denotes the hyperspace of all elements in H (X) intersecting K. In this paper we present some topological properties of the quotient space H (X) / H (X ; K) , going forward in its study in the available literature. In the class of finite graphs, we study the problem of determining conditions on X and K such that C n (X) and C n (X) / C n (X ; K) are homeomorphic, obtaining in this direction some characterizations. [ABSTRACT FROM AUTHOR]

This paper continues a program due to Motegi regarding universal bounds for the number of nonisotopic essential n -punctured tori in the complement of a hyperbolic knot in S 3. For n = 1 , Valdez-Sánchez showed that there are at most five nonisotopic Seifert tori in the exterior of a hyperbolic knot. In this paper, we address the case n = 2. We show that there are at most six nonisotopic, nested, essential 2-holed tori in the complement of every hyperbolic knot. [ABSTRACT FROM AUTHOR]

In many papers, the concept of chaos is associated with positive topological entropy. The points around which chaos is concentrated are also often examined. In this paper we distinguish places of chaos generation and its realization. Moreover, we will consider the possibility of approximation of continuous function by functions having full entropy point x 0 (there is chaos generated by any neighbourhood of it) but simultaneously x 0 is not a focal entropy point (chaos is realized at some distance from the considered point). [ABSTRACT FROM AUTHOR]

For an extended locally convex space (X , τ) , in [12] , the authors studied the finest locally convex topology (flc topology) τ F on X coarser than τ. One can often prove facts about (X , τ) by applying classical locally convex space theory on (X , τ F). This paper employs the flc topology to analyze barreled extended locally convex spaces and establish a version of the Banach-Steinhaus theorem in the extended setting. One of the key results of this paper is the relationship between the barreledness of an extended locally convex space (X , τ) and the barreledness of the associated finest locally convex space (X , τ F). This is achieved by examining the lower semi-continuous seminorms on these spaces. [ABSTRACT FROM AUTHOR]

In the paper we generalize the singular cubical homology theory of digraphs and introduce a collection of cell singular homologies that are parametrized by a natural parameter of the category of digraphs. The natural parameter corresponds to the "length" of a directed line digraph that corresponds to the unit segment in algebraic topology. The introduced earlier singular cubical homology is a particular case of this collection that corresponds to the line digraph with one arrow. We describe relations between these theories for different parameters given by a natural number and study their basic properties. In particular, we prove the functoriality and homotopy invariance of these theories and compare the theory with the path homology theory that was introduced in our previous papers. Then we describe the transfer of the cell singular homology theories to the categories of graphs, quivers, and multigraphs. [ABSTRACT FROM AUTHOR]

This paper is a continuation of our previous paper [10] in which we developed the theory for classifying geometric fibrations of compact, connected, flat n -orbifolds, over a 1-orbifold, up to affine equivalence. In this paper, we apply our theory to classify all the geometric fibrations of compact, connected, flat 3-orbifolds, over a 1-orbifold, up to affine equivalence. [ABSTRACT FROM AUTHOR]

A 4D universe is a 4-dimensional boundary-less connected oriented manifold with every closed 3-manifold (i.e., a 3-dimensional closed connected oriented manifold) embedded. A 4D punctured universe is a 4-dimensional boundary-less connected oriented manifold with the punctured manifold of every closed 3-manifold embedded. Every 4D universe and every 4D punctured universe are open 4-dimensional manifolds. If a closed 3-manifold is considered as a 3D universe, then every 4D spacetime is embedded in every 4D universe and hence every 4D universe is a classifying space for every spacetime. In this paper, it is observed that a full 4D universe is produced by collision modifications between 3-sphere fibers in the 4D spherical shell (i.e., the 3-sphere bundle over the real line) embedded properly in any 5-dimensional open manifold. As a previous result, it was shown that any 4D universe and 4D punctured universe must have infinity on some homological indexes. It is shown in this paper that the second rational homology groups of every 4D universe and every 4D punctured universe are always infinitely generated. [ABSTRACT FROM AUTHOR]

In this paper we study an open problem: Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence? Since every ANR has the homotopy type of a polyhedron, this question is related in an obvious way to the longstanding Borsuk's problem (1967) Is it true that two ANR's homotopy dominating each other have the same homotopy type? The answer is positive for all polyhedra (or ANR's) with polycyclic-by-finite fundamental groups, manifolds, and clearly for 1-dimensional polyhedra. In some previous paper, we proved that this also holds for all 2-dimensional polyhedra with weakly Hopfian (hence Hopfian and co-Hopfian) fundamental groups (a group is weakly Hopfian if it is not isomorphic to a proper retract of itself). In this paper we extend this result to all (G , n) -complexes (where n ≥ 2). A (G , n) -complex, is an n -dimensional finite CW -complex with fundamental group G and all the homotopy groups π r in dimensions 1 < r < n trivial. We also discuss fundamental groups of potential counterexamples. As a corollary we obtain that for the same classes of spaces, the Borsuk's question: Is it true that the homotopy types of two quasi-homeomorphic ANR's are equal? , has positive answer (this question was also asked by S. Ferry in the recent edition of Scottish Book (1981)). [ABSTRACT FROM AUTHOR]

In a previous paper [6] , we obtained, as a consequence of Flyping Theorem due to Menasco and Thislethwaite, that the q -periodicity (q > 2) of an alternating knot can be visualized in an alternating projection as a rotation of the projection sphere. See also [2]. In this paper, we show that the free q -periodicity (q > 2) of an alternating knot can be represented on some alternating projection as a composition of a rotation of order q with some flypes all occurring on the same twisted band diagram of its essential Conway decomposition. Therefore, for an alternating knot to be freely periodic, its essential decomposition must satisfy certain conditions. We show that any free or non-free q -action is some way visible (virtually visible) and give some sufficient criteria to determine from virtually visible projections the existence of a q -action. Finally, we show how the Murasugi decomposition into atoms as initiated in [12] and [13] enables us to determine the visibility type (q , r) of the freely q -periodic alternating knots ((q , r) -lens knots [3]); in fact, we only need to focus on a certain atom of their Murasugi decomposition to deduce their visibility type. [ABSTRACT FROM AUTHOR]

The concept of a knot mosaic was introduced by Lomonaco and Kauffman as a means to construct a quantum knot system. The mosaic number of a given knot K is defined as the minimum integer n that allows the representation of K on an n × n mosaic board. Building upon this, the first author and Nelson extended the knot mosaic system to encompass surface-links through the utilization of marked graph diagrams and established both lower and upper bounds for the mosaic number of the surface-links presented in Yoshikawa's table. In this paper, we establish a mosaic system for immersed surface-links by using singular marked graph diagrams. We also provide the definition and discussion on the mosaic number for immersed surface-links. [ABSTRACT FROM AUTHOR]

In this paper we aim to characterize uniformly continuous real functions and pseudometrics on metric spaces, having uniformly continuous extension. For functions we use a very similar approach as McShane in [7] using moduli of continuity. By doing that we obtain an explicit formula for the extension. We also show that our characterization for functions is equivalent to one proposed in [8] for uniform spaces. We then show that a similar approach can be done for uniformly continuous pseudometrics. To do so we use the notion of chainable metric spaces and intrinsic metrics defined in [9]. A somewhat similar approach has been studied in [6] for normed linear spaces. [ABSTRACT FROM AUTHOR]

In this research paper, we investigate ultra-pseudo metric spaces and prove that a completely regular topological space is homeomorphic to a subspace of a product of ultra-pseudo metric spaces. [ABSTRACT FROM AUTHOR]

This paper offers a novel homotopical characterization of strongly contextual simplicial distributions with binary outcomes, specifically those defined on the cone of a 1-dimensional space. In the sheaf-theoretic framework, such distributions correspond to non-signaling distributions on measurement scenarios where each context contains 2 measurements with binary outcomes. To establish our results, we employ a homotopical approach that includes collapsing measurement spaces and introduce categories associated with simplicial distributions that can detect strong contextuality. [ABSTRACT FROM AUTHOR]

In this paper, we introduce polytopal k -wedge construction and blowdown of a simple polytope and inspect the effect on the retraction sequence of a simple polytope due to the k -wedge construction and blowdown. In relation to these constructions, we introduce the k -wedge and blowdown of a quasitoric orbifold. We compare the torsions in the integral cohomologies of k -wedges and blowdowns of a quasitoric orbifold with the original one. These two constructions provide infinitely many integrally equivariantly formal quasitoric orbifolds from a given one. [ABSTRACT FROM AUTHOR]

Groups with a topology that is in consistent one way or another with the algebraic structure are considered. Well-known classes of groups with a topology are topological, paratopological, semitopological, and quasitopological groups. We also study other ways of matching topology and algebraic structure. The minimum requirement in this paper is that the group is a right semitopological group (such groups are often called right topological groups). We study when a group with a topology is a topological group; research in this direction began with the work of Deane Montgomery and Robert Ellis. (Invariant) semi-neighborhoods of the diagonal are used as a means of study. [ABSTRACT FROM AUTHOR]

The theory of generalized metrizable spaces is an important topic of general topology. This paper is a survey of research methods and achievements on generalized metrizable properties in topological groups and weakly topological groups. We mainly study this kind of properties in topological groups, semitopological groups, paratopological groups, quasitopological groups and free topological groups, and focus on the influence of separation properties, conditions for weakly topological groups to become topological groups, cardinal invariants, weak first-countability, three-space properties and remainders in compactifications on topological groups and related structures. Finally, some unsolved problems in this field are listed for researchers. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of a semi-stratifiable frame as an extension of classical semi-stratifiability and also as the monotonization of perfect frames. We show that stratifiable frames are precisely the monotonically normal semi-stratifiable frames. Moreover, we present an insertion theorem for semi-stratifiable frames in terms of real functions and thereby obtain an insertion theorem for stratifiable frames. [ABSTRACT FROM AUTHOR]

A continuum is a nondegenerate, compact, connected, metric space. A topological property P is invariant under retraction provided that each retract of a continuum having P has P , and P is reversible under retraction by pseudo-deformation if the condition a subcontinuum of a continuum X has P implies that X has P. In this paper, we prove that the absence of R 3 -sets, the absence of R 4 -continua and the absence of s-points are reversible under retractions by pseudo-deformation, and the absence of R 3 -sets and the absence of R 4 -continua are invariant under retractions while the absence of s-points is not. [ABSTRACT FROM AUTHOR]

This article has been retracted: please see Elsevier Policy on Article Withdrawal (https://www.elsevier.com/about/policies/article-withdrawal). This article has been retracted at the request of the Editors-in-Chief after receiving a complaint about citation issues. The editors solicited further independent reviews which indicated that the statements in this paper substantially overlap with the PhD thesis of Z. Furdzik [O własnościach pewnych rozkładów przestrzeni topologicznych na iloczyny kartezjańskie, 1968 (unpublished, in Polish), Institute of Mathematics of the Polish Academy of Sciences, Warsaw], without proper citation. [ABSTRACT FROM AUTHOR]

In this paper, the notion of an original point for a topological space is introduced. Then characterizations of some spaces such as stratifiable spaces and MCP-spaces in terms of set-valued maps with values in F (Y) , where Y has an original point, are presented. [ABSTRACT FROM AUTHOR]

This article has been retracted: please see Elsevier Policy on Article Withdrawal (https://www.elsevier.com/about/policies/article-withdrawal). This article has been retracted at the request of the Editors-in-Chief after receiving a complaint about citation issues. The editors solicited further independent reviews which indicated that the statements in this paper substantially overlap with the PhD thesis of Z. Furdzik [O własnościach pewnych rozkładów przestrzeni topologicznych na iloczyny kartezjańskie, 1968 (unpublished, in Polish), Institute of Mathematics of the Polish Academy of Sciences, Warsaw], without proper citation. [ABSTRACT FROM AUTHOR]

In this paper we continue researches deal with increasing sequence { T m } of T 0 -topologies on the set of positive integers focusing on the properties of arithmetic progressions in topologies T m. We characterize the closures of arithmetic progressions in all topologies T m on N. Additionally, we present the characterization of the closures of arithmetic progressions in the common division topology T on N. Moreover, for each m ∈ N we characterize regular open arithmetic progressions in (N , T m) and we examine which of these spaces are semiregular. [ABSTRACT FROM AUTHOR]

In the paper, we study a digital topological complexity of a digital map and its properties. Firstly, we discuss a strong digital homotopy which allows iterative algorithms based on our previous work. As a generalization of the topological complexity in terms of the strong digital homotopy (we call it strong digital topological complexity), we next study the strong digital f -sectional category of a strong digital fibration. Then we investigate estimates of the upper and lower bounds for the strong digital topological complexity of digital maps. We also reveal the difference between the strong digital topological complexity and the ordinary digital ones. It has shown that the strong digital topological complexity is more similar to the classical continuous case than the ordinary digital ones. Moreover, arising from practical considerations in robotics, we consider the naive digital topological complexity of digital maps. [ABSTRACT FROM AUTHOR]

A topological space X is called almost discrete, if it has precisely one nonisolated point. In this paper, we get that for a countable product X = ∏ X i of almost discrete spaces X i the space C p (X) of all continuous real-valued functions with the topology of pointwise convergence is a μ -space if, and only if, X is a weak q -space if, and only if, t (X) = ω if, and only if, X is functionally generated by the family of all its countable subspaces. This result makes it possible to solve Archangel'skii's problem on the product of Grothendieck spaces. It is proved that in the model of ZFC , obtained by adding one Cohen real, there are Grothendieck spaces X and Y such that X × Y is not weakly Grothendieck space. In (P F A) : the product of any countable family almost discrete Grothendieck spaces is a Grothendieck space. [ABSTRACT FROM AUTHOR]

Let f : X → Y be a continuous surjection of compact Hausdorff spaces. By f ⁎ : M (X) → M (Y) , μ ↦ μ ∘ f − 1 and 2 f : 2 X → 2 Y , A ↦ f [ A ] we denote the induced continuous surjections on the probability measure spaces and hyperspaces, respectively. In this paper we mainly show the following facts: (1) If f ⁎ is semi-open, then f is semi-open. (2) If f is semi-open densely open, then f ⁎ is semi-open densely open. (3) f is open iff 2 f is open. (4) f is semi-open iff 2 f is semi-open. (5) f is irreducible iff 2 f is irreducible. [ABSTRACT FROM AUTHOR]

It is well known that the Hausdorff dimension and box dimension of a Bedford-McMullen carpet coincide if and only if the carpet has uniform horizontal fibers. Yang et al. [17] gave a complete Lipschitz classification of such carpets which are totally disconnected and possessing vacant rows. In this paper, we show that if we replace the uniform horizontal fibers condition by a locally uniform horizontal fibers condition, the conclusions of Yang et al. still hold. [ABSTRACT FROM AUTHOR]

We introduce the selection principles H-star-Lindelöf, weakly H-star-Lindelöf and almost H-star-Lindelöf and we provide some propositions and relationships with other known properties in literature. Furthermore, in this paper we characterize the above selection principles in hyperspaces endowed with the hit-and-miss topology τ Δ , by using a new property SS Π Δ (Λ) , fin ⋆ (C Δ (Λ) , B) , for the corresponding choose of the family B. [ABSTRACT FROM AUTHOR]

Abstract In this article we study the higher topological complexity TC r (X) in the case when X is an aspherical space, X = K (π , 1) and r ≥ 2. We give a characterisation of TC r (K (π , 1)) in terms of classifying spaces for equivariant Bredon cohomology. Our recent paper [7] , joint with M. Grant and G. Lupton, treats the special case r = 2. We also obtain in this paper useful lower bounds for TC r (π) in terms of cohomological dimension of subgroups of π × π × ... × π (r times) with certain properties. As an illustration of the main technique we find the higher topological complexity of Higman's group. We also apply our method to obtain a lower bound for the higher topological complexity of the right angled Artin (RAA) groups, which, as was established in [16] by a different method (in a more general situation), coincides with the precise value. We finish the paper by a discussion of the TC -generating function ∑ r = 1 ∞ TC r + 1 (X) x r encoding the values of the higher topological complexity TC r (X) for all values of r. We show that in many examples (including the case when X = K (H , 1) with H being a RAA group) the TC -generating function is a rational function of the form P (x) (1 − x) 2 where P (x) is an integer polynomial with P (1) = cat (X). [ABSTRACT FROM AUTHOR]

Abstract Based on the notions of T. Banakh's strict Pytkeev networks and A.V. Arhangel'skiı̌'s sensor families, strict Pytkeev networks with sensors are introduced in this paper. A family P of subsets of a topological space X is called a strict Pytkeev network with sensors (abbr. an sp-network) if, for each x ∈ U ∩ A ‾ with U open and A subset in X , there is a set P ∈ P such that x ∈ P ⊂ U and x ∈ P ∩ A ‾. In present paper, we discuss certain relationship and operations among spaces defined by special Pytkeev networks, study spaces with a point-countable sp -network and spaces with a σ -closure-preserving sp -network, and detect some applications of sp -networks in topological groups. The following results are obtained: (1) Every sp -network is preserved by a continuous pseudo-open mapping. (2) Every k -space with a point-countable sp -network coincides with a continuous pseudo-open s -image of a metric space. (3) Every regular feebly compact space with a point-countable sp -network has a point-countable base. (4) A regular space has a countable sp -network if and only if it is separable and has a point-countable sp -network. (5) A topological space is stratifiable if and only if it is a regular space with a σ -closure-preserving sp -network. (6) A regular space with a σ -locally finite sp -network has a σ -discrete sp -network. (7) A topological group is metrizable if it has countable sp -character. (8) There is a non-Fréchet-Urysohn sequential topological group with a countable strict Pytkeev network, which give a negative answer to a question posed by A.V. Arhangel'skiı̌ [1]. [ABSTRACT FROM AUTHOR]

Abstract Regarding the calculation of homotopy classes of comultiplications on a wedge of spheres, the more spheres that appear in the wedge, the more complicated proofs and computations become. In this paper, we concentrate on the calculations of the comultiplications and the associative and commutative comultiplications on the k -fold wedge of spheres ∨ i = 1 k S n i for sufficiently large values of k as a generalization of the papers [3,4]. In particular, we observe extraordinary regularities and patterns on the conditions of associative and commutative comultiplications on the k -fold wedge of spheres. [ABSTRACT FROM AUTHOR]

Abstract On the category of compact metric spaces an exact homology theory was defined and its relation to the Vietoris homology theory was studied by N. Steenrod [11]. In particular, the homomorphism from the Steenrod homology groups to the Vietoris homology groups was defined and it was shown that the kernel of the given homomorphism are homological groups, which was called weak homology groups [11] , [3]. The Steenrod homology theory on the category of compact metric pairs was axiomatically described by J. Milnor. In [10] the uniqueness theorem is proved using the Eilenberg–Steenrod axioms and as well as relative homeomorphism and clusters axioms. J. Milnor constructed the homology theory on the category T o p C 2 of compact Hausdorff pairs and proved that on the given category it satisfies nine axioms — the Eilenberg–Steenrod, relative homeomorphism and cluster axioms (see theorem 5 in [10]). Besides, using the construction of weak homology theory, J. Milnor proved that constructed homology theory satisfies partial continuity property on the subcategory T o p C M 2 (see theorem 4 in [10]) and the universal coefficient formula on the category T o p C 2 (see Lemma 5 in [10]). On the category of compact Hausdorff pairs, different axiomatic systems were proposed by N. Berikashvili [1] , [2] , H. Inasaridze and L. Mdzinarishvili [7] , L. Mdzinarishvili [9] and H. Inasaridze [6] , but there was not studied any connection between them. The paper studies this very problem. In particular, in the paper it is proved that any homology theory in Inasaridze sense is the homology theory in the Berikashvili sense, which itself is the homology theory in the Mdzinarishvili sense. On the other hand, it is shown that if a homology theory in the Mdzinarishvili sense is exact functor of the second argument, then it is the homology in the Inasaridze sense. [ABSTRACT FROM AUTHOR]

For a Tychonoff space X , we denote by C p ( X ) the space of all real-valued continuous functions on X with the topology of pointwise convergence. In this paper we continue to study different selectors for sequences of dense sets of C p ( X ) started to study in the paper [14] . A set A ⊆ C p ( X ) will be called 1 -dense in C p ( X ) , if for each x ∈ X and an open set W in R there is f ∈ A such that f ( x ) ∈ W . We give the characterizations of selection principles S 1 ( A , A ) , S f i n ( A , A ) and S 1 ( S , A ) where • A — the family of 1-dense subsets of C p ( X ) ; • S — the family of sequentially dense subsets of C p ( X ) , through the selection principles of a space X . In particular, we give the functional characterizations of the Rothberger and Menger properties. [ABSTRACT FROM AUTHOR]

For a Tychonoff space X , C p ( X ) is the space of all real-valued continuous functions with the topology of pointwise convergence. A subset A ⊂ X is said to be sequentially dense in X if every point of X is the limit of a convergent sequence in A . In this paper, the following 8 properties for C p ( X ) are considered. S 1 ( S , S ) ⇒ S f i n ( S , S ) ⇒ S 1 ( S , D ) ⇒ S f i n ( S , D ) ⇑ ⇑ ⇑ ⇑ S 1 ( D , S ) ⇒ S f i n ( D , S ) ⇒ S 1 ( D , D ) ⇒ S f i n ( D , D ) For example, a space X satisfies S 1 ( D , S ) (resp., S f i n ( D , S ) ) if whenever { D n : n ∈ N } is a sequence of dense subsets of X , one can take points x n ∈ D n (resp., finite F n ⊂ D n ) such that { x n : n ∈ N } (resp., ⋃ { F n : n ∈ N } ) is sequentially dense in X . Other properties are defined similarly. S 1 ( D , D ) (= R -separability) and S f i n ( D , D ) (= M -separability) for C p ( X ) were already investigated by several authors. In this paper, we have gave characterizations for C p ( X ) to satisfy other 6 properties above. [ABSTRACT FROM AUTHOR]

In this paper, we give definitions of three kinds of minimal charts, and we investigate properties of minimal charts and establish fundamental theorems characterizing minimal charts. To classify charts with two or three crossings we use the fundamental theorems. In the future paper, we give an enumeration of the charts with two crossings. [ABSTRACT FROM AUTHOR]

We analyze the sequence obtained by consecutive applications of the Cantor–Bendixson derivative for a noncommutative scattered C ⁎ -algebra A , using the ideal I A t ( A ) generated by the minimal projections of A . With its help, we present some fundamental results concerning scattered C ⁎ -algebras, in a manner parallel to the commutative case of scattered compact or locally compact Hausdorff spaces and superatomic Boolean algebras. It also allows us to formulate problems which have motivated the “cardinal sequences” programme in the classical topology, in the noncommutative context. This leads to some new constructions of noncommutative scattered C ⁎ -algebras and new open problems. In particular, we construct a type I C ⁎ -algebra which is the inductive limit of stable ideals A α , along an uncountable limit ordinal λ , such that A α + 1 / A α is ⁎-isomorphic to the algebra of all compact operators on a separable Hilbert space and A α + 1 is σ -unital and stable for each α < λ , but A is not stable and where all ideals of A are of the form A α . In particular, A is a nonseparable C ⁎ -algebra with no ideal which is maximal among the stable ideals. This answers a question of M. Rørdam in the nonseparable case. All the above C ⁎ -algebras A α s and A satisfy the following version of the definition of an AF algebra: any finite subset can be approximated from a finite-dimensional subalgebra. Two more complex constructions based on the language developed in this paper are presented in separate papers [21,22] . [ABSTRACT FROM AUTHOR]

Sapirovskii [16] proved that | X | ≤ π χ ( X ) c ( X ) ψ ( X ) , for a regular space X . We introduce the θ - pseudocharacter of a Urysohn space X , denoted by ψ θ ( X ) , and prove that the previous inequality holds for Urysohn spaces replacing the bounds on cellularity c ( X ) ≤ κ and on pseudocharacter ψ ( X ) ≤ κ with a bound on Urysohn cellularity U c ( X ) ≤ κ (which is a weaker condition because U c ( X ) ≤ c ( X ) ) and on θ -pseudocharacter ψ θ ( X ) ≤ κ respectively (note that in general ψ ( ⋅ ) ≤ ψ θ ( ⋅ ) and in the class of regular spaces ψ ( ⋅ ) = ψ θ ( ⋅ ) ). Further, in [6] the authors generalized the Dissanayake and Willard's inequality: | X | ≤ 2 a L c ( X ) χ ( X ) , for Hausdorff spaces X [21] , in the class of n -Hausdorff spaces and de Groot's result: | X | ≤ 2 h L ( X ) , for Hausdorff spaces [10] , in the class of T 1 spaces (see Theorems 2.22 and 2.23 in [6] ). In this paper we restate Theorem 2.22 in [6] in the class of n -Urysohn spaces and give a variation of Theorem 2.23 in [6] using new cardinal functions, denoted by U W ( X ) , ψ w θ ( X ) , θ - a L ( X ) , h θ - a L ( X ) , θ - a L c ( X ) and θ - a L θ ( X ) . In [5] the authors introduced the Hausdorff point separating weight of a space X denoted by H p s w ( X ) and proved a Hausdorff version of Charlesworth's inequality | X | ≤ p s w ( X ) L ( X ) ψ ( X ) [7] . In this paper, we introduce the Urysohn point separating weight of a space X , denoted by U p s w ( X ) , and prove that | X | ≤ U p s w ( X ) θ - a L c ( X ) ψ ( X ) , for a Urysohn space X . [ABSTRACT FROM AUTHOR]

This paper is devoted to studies of I wQN-spaces and some of their cardinal characteristics. Recently, Šupina in [32] proved that I is not a weak P-ideal if and only if any topological space is an I QN-space. Moreover, under p = c he constructed a maximal ideal I (which is not a weak P-ideal) for which the notions of I QN-space and QN-space do not coincide. In this paper we show that, consistently, there is an ideal I (which is not a weak P-ideal) for which the notions of I wQN-space and wQN-space do not coincide. This is a partial solution to [6, Problem 3.7] . We also prove that for this ideal the ideal version of Scheepers Conjecture does not hold (this is the first known example of such weak P-ideal). We obtain a strictly combinatorial characterization of non ( I wQN-space ) similar to the one given in [32] by Šupina in the case of non ( I QN-space ) . We calculate non ( I QN-space ) and non ( I wQN-space ) for some weak P-ideals. Namely, we show that b ≤ non ( I QN-space ) ≤ non ( I wQN-space ) ≤ d for every weak P-ideal I and that non ( I QN-space ) = non ( I wQN-space ) = b for every F σ ideal I as well as for every analytic P-ideal I generated by an unbounded submeasure (this establishes some new bounds for b ( I , I , Fin ) introduced in [31] ). As a consequence, we obtain some bounds for add ( I QN-space ) . In particular, we get add ( I QN-space ) = b for analytic P-ideals I generated by unbounded submeasures. By a result of Bukovský, Das and Šupina from [6] it is known that in the case of tall ideals I the notions of I QN-space ( I wQN-space) and QN-space (wQN-space) cannot be distinguished. Answering [6, Problem 3.2] , we prove that if I is a tall ideal and X is a topological space of cardinality less than co v ⁎ ( I ) , then X is an I wQN-space if and only if it is a wQN-space. [ABSTRACT FROM AUTHOR]

Let P k be the graded polynomial algebra F 2 [ x 1 , x 2 , ... , x k ] over the prime field of two elements, F 2 , with the degree of each x i being 1. We study the hit problem , set up by Frank Peterson, of finding a minimal set of generators for P k as a module over the mod-2 Steenrod algebra, A. In this paper, we explicitly determine a minimal set of A -generators for P 5 in the case of the generic degrees n = 2 d + 1 − 1 and n = 2 d + 1 − 2 for all d ⩾ 6. [ABSTRACT FROM AUTHOR]

Let C (X) be the hyperspace of all subcontinua of a continuum X topologized by Hausdorff metric. For a non-empty closed subset A of a continuum X , consider the following properties: A is a strong non-cut subset, non-block subset, weak non-block subset, shore subset, not a strong center, and non-cut subset of X. The aim of this paper is to study the conditions under which an ordered arc from a singleton to a proper subcontinuum of a continuum X has one of these properties in C (X). [ABSTRACT FROM AUTHOR]

Let (X , d) be a metric space and f 1 , ∞ = { f n } i = 0 ∞ be a sequence of continuous maps f n : X → X such that (f n) converges uniformly to a continuous map f. We investigate which conditions ensure that the transitivity of functions f n or the transitivity of the nonautonomous system (X , f 1 , ∞) is inherited to the limit function f and vice versa. Such problem has been studied for instance by A. Fedeli, A. Le Donne or J. Li who give different sufficient condition for inheriting of transitivity from f n to f. In this paper we give a survey of known result relating to this problem and prove new results concerning transitivity. [ABSTRACT FROM AUTHOR]

In this paper, we study the notions of F -sensitivity, multi- F -sensitivity and (F 1 , F 2) -sensitivity in dynamical systems defined on Hausdorff uniform spaces. It is shown that how these notions carry over to hyperspatial and product dynamical systems, and vice versa. We also obtain some sufficient conditions for a dynamical system to be F -sensitive. Some examples showing the necessity of the conditions taken in our results are also presented. • The notions of F -sensitivity, multi- F -sensitivity and (F 1 , F 2) -sensitivity have been studied in systems defined on Hausdorff uniform spaces. • Sufficient conditions for F -sensitivity using F -transitivity have been provided. • We prove that a dynamical system (X , T) is multi- F -sensitive if and only if the system (C (X) , C T) is multi- F -sensitive. • We prove that if F 1 is a filter and F 2 is an ultrafilter, then the (F 1 , F 2) -sensitivity of (X , T) implies that of (C (X) , C T). • Moreover, all the aforementioned notions have been studied on arbitrary products of dynamical systems. [ABSTRACT FROM AUTHOR]

Suppose that (X , d) is a complete metric space and M is a nonempty closed proper subset of X. Let μ M be the Gromov hyperbolic metric on X ∖ M defined by Z. Ibragimov. In this paper, we show that there exists a natural quasisymmetric correspondence between the Gromov boundary of (X ∖ M , μ M) and its metric boundary ∂ M. [ABSTRACT FROM AUTHOR]

In this paper, we examine the relations between classical homotopy groups, digital fundamental groups of L. Boxer [6] , and digital homotopy classes on the digital wedge products of pointed digital diamond (or square) circles. We demonstrate that the classical homotopy groups of pointed functions from a wedge product of homotopy simple closed curves to itself may differ completely from the digital homotopy monoids consisting of the classes of NP u pointed homotopic functions from the digital wedge product of pointed digital diamond (or square) circles to itself under the κ -adjacency relations on Z 2 for κ = 4 , 8 and u ∈ { 1 , 2 }. [ABSTRACT FROM AUTHOR]

We introduce the notion of a system X = (X a , p a b , (A , ⪯)) in a category C where (A , ⪯) is a pre-ordered directed set, for each a ∈ A , X a ∈ Obj (C) , and whenever a ⪯ b there is defined a unique C -morphism p a b : X b → X a. If a system admits certain commutative diagrams of a type defined in this paper, then it is called a delay-inverse system. In these systems the usual commutative diagrams p a c = p a b p b c when a ⪯ b ⪯ c of an inverse system might fail. However, commutativity is always recovered "after some delay." The authors encountered, accidentally, a profuse class of naturally occurring delay-inverse systems that are not inverse systems in their study of certain classes of Čech systems and their relation to approximate systems in the sense of Mardešić and Watanabe. [ABSTRACT FROM AUTHOR]

A linear constraint system is specified by linear equations over the group Z d of integers modulo d. Their operator solutions play an important role in the study of quantum contextuality and non-local games. In this paper, we use the theory of simplicial sets to develop a framework for studying operator solutions of linear systems. Our approach refines the well-known group-theoretical approach based on solution groups by identifying these groups as algebraic invariants closely related to the fundamental group of a space. In this respect, our approach also makes a connection to the earlier homotopical approach based on cell complexes. Within our framework, we introduce a new class of linear systems that come from simplicial sets and show that any linear system can be reduced to one of that form. Then we specialize in linear systems that are associated with groups. We provide significant evidence for a conjecture stating that for odd d every linear system admitting a solution in a group admits a solution in Z d. [ABSTRACT FROM AUTHOR]

A Boolean algebra A equipped with a (finitely-additive) positive probability measure m can be turned into a metric space (A , d m) , where d m (a , b) = m ((a ∧ ¬ b) ∨ (¬ a ∧ b)) , for any a , b ∈ A , sometimes referred to as metric Boolean algebra. In this paper, we study under which conditions the space of atoms of a finite metric Boolean algebra can be isometrically embedded in R N (for a certain N) equipped with the Euclidean metric. In particular, we characterize the topology of the positive measures over a finite algebra A such that the metric space (At (A) , d m) embeds isometrically in R N (with the Euclidean metric). [ABSTRACT FROM AUTHOR]

In this paper we investigate the basic properties of those dynamical systems which are positively expansive on a dense set containing all eventually periodic points and show how the dynamical behavior of these systems fits within the theoretical framework of usual (positive) expansiveness. [ABSTRACT FROM AUTHOR]

Let X be a finite CW-complex having mod p cohomology isomorphic to a wedge of three spheres S n ∨ S m ∨ S l , 1 ≤ n ≤ m ≤ l. The aim of this paper is to determine the fixed point sets of actions of the cyclic group of prime order on X. We also classify the orbit spaces of free actions of G = Z p , p a prime or G = S d , d = 1 or 3, on X and derive the Borsuk-Ulam type results. [ABSTRACT FROM AUTHOR]