In a recent article, Cumplido and Paris studied the question of commensurability between Artin groups of spherical type. Their analysis left six cases undecided, for the following pairs of Artin groups: (F4,D4), (H4,D4), (F4,H4), (E6,D6), (E7,D7), and (E8,D8). In this note we resolve the first two of these cases, namely, we show that the Artin groups of types F4 and H4 are not commensurable with that

In this paper, a partial order and its basic properties are exploited in a fuzzy quasi-metric space in the sense of George and Veeramani (A. Gregori, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst. 64 (1994) 395–399). Based on this partial order, Caristi's fixed point theorem, Ekeland's variational principle and Takahashi's maximization theorem are extended to fuzzy quasi-metric

In this paper, we investigate some versions of d-space, well-filtered space and Rudin space concerning various countability properties. It is proved that every T0 space with a first-countable sobrification is an ω-Rudin space and every first-countable T0 space is well-filtered determined. Therefore, every ω-well-filtered space with a first-countable sobrification is sober. It is also shown that every

Some sequences S of measurable sets tending to zero generate density-like topologies TS on the real line finer than the natural topology Tnat. For a topology TS we consider the metric space CS,S of S-continuous functions f:(R,TS)→(R,TS), and we consider the metric spaces Cnat,nat and C+ of continuous functions and of right continuous functions, respectively, all spaces endowed with the supremum metric

G. Debs and J. Saint Raymond in 2009 defined the Borel separation rank of an analytic ideal I (rk(I)) as minimal ordinal α<ω1 such that there is S∈Σ1+α0 with I⊆S and I⋆∩S=∅, where I⋆ is the filter dual to the ideal I. Moreover, they introduced ideals Finα, for all α<ω1, and conjectured that rk(I)≥α if and only if I contains an isomorphic copy of Finα (Finα⊑I). To define Finα in the case of limit ordinals

We discuss the monotonicity problem for the amenable category of closed manifolds under maps of non-zero degree. In dimension three this leads to a variation on a recent theorem of Capovilla, Löh and Moraschini [1].

We show that the Menger curve can be represented as a set-valued inverse limit of intervals [0,1] with a single bonding function. The graph of the function is homeomorphic to the Sierpiński carpet. This answers a question by M. Hiraki and H. Kato.

In this paper we develop a new approach to the study of uncountable fundamental groups by using Hurewicz fibrations with the unique path-lifting property (lifting spaces for short) as a replacement for covering spaces. In particular, we consider the inverse limit of a sequence of covering spaces of X. It is known that the path-connectivity of the inverse limit can be expressed by means of the derived

Using the notions of Topological dynamics, H. Furstenberg defined central sets and proved the Central Sets Theorem. Later V. Bergelson and N. Hindman characterized central sets in terms of algebra of the Stone-Čech Compactification of discrete semigroup. They found that central sets are the members of minimal idempotents of βS, the Stone-Čech Compactification of a semigroup (S,⋅). We know that any

In work from 2004, Cimasoni gave a geometric computation of the multivariable Conway potential function in terms of a generalization of a Seifert surface for a link called a C-complex [2]. These surfaces were introduced by Cooper [9], [10] for 2-component links and were used to compute Alexander invariants, signatures and nullities. Cooper presents a sequence of geometric moves which relate any two

We establish some results about countable products of spaces that are either consonant or whose complement in some compactification has the Menger Property. We use our results to answer some questions of A.Bouziad.

The aim of this paper is to consider questions concerning the possible maximum cardinality of various separable pseudoradial (in short: SP) spaces. The most intriguing question here is if there is in ZFC a regular (or just Hausdorff) SP space of cardinality >c. While this question is left open, we establish a number of non-trivial results that we list below. • It is consistent with MA+c=ℵ2 that there

Given a T0 paratopological group G and a class C of continuous homomorphisms of paratopological groups, we define the C-semicompletion C[G) and C-completion C[G] of the group G that contain G as a dense subgroup, satisfy the T0-separation axiom and have certain universality properties. For special classes C, we present some necessary and sufficient conditions on G in order that the (semi)completions

A betweenness structure on a set X is a ternary relation [⋅,⋅,⋅]⊆X3 that captures a rudimentary notion of one point of X lying between two others. The interval [a,b] is the set of all points lying between a and b, and a subset C of X is convex if [a,b]⊆C whenever a,b∈C. The span of a set A is the union of all intervals [a,b], where a,b∈A; by iterating the span operator countably many times, we obtain

We denote by Cs(X) the set C(X) of real-valued continuous functions defined on X endowed with the topology of the uniform convergence on the closed separable subspaces of X. In this paper we continue the study of Cs(X) initiated in Pseudouniform topologies on C(X) given by ideals (Pichardo-Mendoza et al. (2013) [7]). We prove that Cs(X) is a k-space if and only if Cs(X) is metrizable, and that compactness

For a topologically complete space X and a family of closed covers A of X satisfying a “local refinement condition” and a “completeness condition,” we give a construction of an inverse system NA of simplicial complexes and simplicial bonding maps such that the limit space N∞=lim←NA is homotopy equivalent to X. A connection with cell structures [2], [3] is discussed.

Results in this paper concern finding a cardinal number κ, as small as possible, such that every uniformly or proximally continuous mapping from a given subspace X of a product ∏IXi into some A factorizes via a subspace of ∏JXi with |J|≤κ. In some cases we substantially improve Vidossich theorem from [12]. Influenced by investigating locally co-presentable categories in [8], also cardinals κ are found

The aim of this paper is to study the class of spaces with the FNS property and the π-FNS property. We prove that compact spaces with the FNS property for some base consisting of co-zero sets are openly generated spaces and spaces with the π-FNS property are skeletally generated spaces. We give examples of families without the Freese–Nation property.

Using the classical technique of condensation of singularities, we prove that, for every zero-dimensional, complete separable metric space G, there exists a Suslinian, chainable metric continuum whose set of end points is homeomorphic to G. This answers a question posed by R. Adikari and W. Lewis in [Houston J. Math. 45 (2019), no. 2, pp. 609–624].

Let X be a set and A⊆P(X) be a family closed under finite intersections such that ∅,X∈A. If ψ=o,ω,γ, then Ψ(A) is the family of those ψ-covers U for which U⊆A. In [3], properties (Ψ0) of a family F⊆XR of real functions have been introduced. The main result of the paper Theorem 4.1 reads as follows: if Φ=Ω,Γ and Ψ=O,Ω,Γ, then for any pair 〈Φ,Ψ〉 different from 〈Ω,O〉, X has the covering property S1(Φ(A)

The Golomb space (resp. the Kirch space) is the set N of positive integers endowed with the topology generated by the base consisting of arithmetic progressions a+bN0={a+bn:n≥0} where a,b∈N and b is a (square-free) number, coprime with a. It is known that the Golomb space (resp. the Kirch space) is connected (and locally connected). By a recent result of Banakh, Spirito and Turek, the Golomb space

We construct a continuum X⊆R3 such that there exists a continuous surjection f from X onto X with the inverse image of every point of size continuum and such that for no topological space T whose every point has a nontrivial neighbourhood there exists a surjection from X onto X×T. In particular, f cannot be a composition of a surjection from X onto X×T and the projection onto X, where T is of size

Given n>1, there exists a mutually aposyndetic continuum Σ satisfying the following two conditions: (1) No nondegenerate subcontinuum of Σ is metrizable; and (2) Σ admits a monotone map onto an n-sphere such that every point inverse has 2c points.

The inequality |X|≤2χ(X) has been proved to be true for Lindelöf spaces (Arhangel'skiĭ, 1969), H-closed spaces (Dow-Porter, 1982) and ccc spaces (Hajnal-Juhász 1967), by quite different arguments. We present a common extension of all these properties which allows us to give a unified proof of these three theorems.

For a topological space X we propose to call a subset S⊂X free in X if it admits a well-ordering that turns it into a free sequence in X. The well-known cardinal function F(X) is then definable as sup⁡{|S|:S is free in X} and will be called the free set number of X. We prove several new inequalities involving F(X) and F(Xδ), where Xδ is the Gδ-modification of X: • L(X)≤22F(X) if X is T2 and L(X)≤2F(X)

We determine a nice simple formula for the largest Euclidean space for which there is an orientable n-manifold with a nonimmersion detected by Stiefel-Whitney classes. For Spin manifolds, we prove the analogue of the upper bound and establish the complete answer for n≤23 and 32≤n≤33. Results similar to many of these were obtained some 50 years ago, but in a much less tractable form. The sharp results

We shall show that for every natural number n, every continuous injection from the product of n-many connected nowhere separable linearly ordered spaces into such a product is coordinate-wise. This result was announced by the author in [1].

In this paper we connect selection principles on a topological space to corresponding selection principles on one of its hyperspaces. We unify techniques and generalize theorems from the known results about selection principles for common hyperspace constructions. This includes results of Lj.D.R. Kočinac, Z. Li, and others. We use selection games to generalize selection principles and we work with

We present proofs of Hurewicz's and Pawlikowski's Theorems concerning topological games, but in the context of lattices, in which one cannot argue with points.

The classical Jordan curve theorem for digital curves asserts that the Jordan curve theorem remains valid in the Khalimsky plane. Since the Khalimsky plane is a quotient space of R2 induced by a tiling of squares, it is natural to ask for which other tilings of the plane it is possible to obtain a similar result. In this paper we prove a Jordan curve theorem which is valid for every locally finite

It is shown that the family of all homogeneous continua in the hyperspace of all subcontinua of any finite-dimensional Euclidean cube or the Hilbert cube is an analytic subspace of the hyperspace which contains a topological copy of the linear space c0={(xk)∈Rω:lim⁡xk=0} as a closed subset.

In this paper, we introduced three uniformities VGl, VGr and VG which are induced in a natural way on a strongly topological gyrogroup (G,⊕,τ). It is mainly proved that (1) each of the three uniformities is compatible with G; (2) if H is a subgyrogroup of G, then VG,Hl=VHl, VG,Hr=VHr and VG,H=VH; (3) if {(Gi,⊕i,τi):i∈I} is a family of strongly topological gyrogroups, then VGl=∏i∈IVGil,VGr=∏i∈IVGir

In this paper, we study Dehn colorings for spatial graphs, and give a family of spatial graph invariants that are called vertex-weight invariants. We give some examples of spatial graphs that can be distinguished by a vertex-weight invariant, whereas distinguished by neither their constituent links nor the number of Dehn colorings.

We establish that any Hausdorff discretely κ-Lindelöf space X must be κ-Lindelöf if t(X)≤κ. Besides, in κ-Lindelöf spaces whose tightness does not exceed κ, the property of having pseudocharacter ≤κ must be discretely reflexive. Under the hypothesis c<ωω we prove that countable pseudocharacter is discretely reflexive in Lindelöf Σ-spaces. If X is a Tychonoff space and ψ(D‾)≤κ for any discrete set D⊂Cp(X)

A group G admitting a cyclic presentation P=Pn(w) determines E=G⋊θCn called the shift extension of G. A spherical picture over a presentation for E lifts to one over P, and here it is shown how this picture determines a Heegaard diagram for a 3-manifold inducing P analogous to how spherical diagrams are associated with face pairings. The method is demonstrated with the groups of type Z, an infinite

We explore the connections between selection games on Hausdorff spaces and their corresponding Vietoris space of compact subsets. These considerations offer a similar relationship as the well-known relationship between ω-covers of X and ordinary open covers of the finite powers of X. The primary utility of this method is to establish similar relationships with k-covers and the Vietoris space of compact

Bal and Bhowmik [1] introduced two new star selection principles U1⁎(A,B) and Ufin⁎(A,B). Song and Xuan [17] gave an example of a Tychonoff space X which has the property U1⁎(O,O) having a regular-closed subset that does not have the property U1⁎(O,O), where O denotes the collection of all open covers of a space X. In this paper, we show that there exists a Tychonoff space with the property U1⁎(O,O)

In this paper, we continue the study of the cancellation problem for the cartesian product of partially ordered sets (posets for short) and of T0-spaces. We first give a class of partially ordered sets satisfying the cancellation law which is different from that investigated by Banaschewski and Lowen, and then give its corresponding topological version.

Let ϕ:X×R→X be a continuous flow on a compact metric space (X,d). In this article we constructively prove the existence of a continuous Lyapunov function for ϕ which is strictly decreasing outside SCRd(ϕ). Such a result generalizes Conley's Fundamental Theorem of Dynamical Systems for the strong chain recurrent set.

This paper is devoted to the proof of existence of q-periodic alternating projections of prime alternating q-periodic knots. The main tool is the Menasco-Thistlethwaite's Flyping Theorem. Let K be an oriented prime alternating knot that is q-periodic with q≥3, i.e. that admits a rotation of order q as a symmetry. Then K has an alternating projection Π(K) such that the rotational symmetry of K is visualized

A topological group G with |G|>1 is called d-independent if for every subgroup S of G with |S|<2ω, one can find a countable dense subgroup H of G such that S∩H={e}. Therefore, d-independent groups are separable and have cardinality at least 2ω. Our main result is a purely algebraic characterization of d-independence in the class of compact metrizable abelian groups. We prove that a compact metrizable

Given a metric continuum X, let C(X) and F1(X) be the hyperspaces of subcontinua and the one-point sets of X, respectively. Let D(X) be the collection of all regular subsets in X belonging to C(X) and let M(X) be all the subcontinua of X with empty interior. In the first part of this paper we are interested in the problem to classify continua that satisfies one of the following equalities: F1(X)=M(X)

Let G be a locally compact amenable group, TLIM(G) the topological left-invariant means on G, and TLIM0(G) the limit points of Følner-nets. I show that TLIM0(G)=TLIM(G) unless G is σ-compact non-unimodular, in which case TLIM0(G)≠TLIM(G). This improves a 1970 result of Chou and a 2009 result of Hindman and Strauss. I consider the analogous problem for the non-topological left-invariant means, and give

In this paper, we study affine manifolds endowed with linear foliations. These are foliations defined by vector subspaces invariant by the linear holonomy. We show that an n-dimensional compact, complete, and oriented affine manifold endowed with a codimension 1 linear foliation F is homeomorphic to the n-dimensional torus if the leaves of F are simply connected. Let (M,∇M) be a 3-dimensional compact

In this paper, we introduce some selection principles, which are motivated by star and strong star Rothberger and star and strong star Menger principles. Theorems to characterize the star and strong star Rothberger-type properties in spaces using the family CΔ(Λ) of cΔ(Λ)-covers of a space X and the family D of dense subsets in hit-and-miss hyperspaces are proved. In a similar way are presented the

We establish universality and ultra-homogeneity of (U,uGH), the collection of all compact ultrametric spaces endowed with the so-called Gromov-Hausdorff ultrametric. This result also gives rise to a novel construction of the so-called R-Urysohn universal ultrametric space for each countable subset R⊂R≥0 containing 0.

We deal with the question of Masayoshi Hata: is every Peano continuum a topological fractal? A compact space X is a topological fractal if there exists F a finite family of self-maps on X such that X=⋃f∈Ff(X) and for every open cover U of X there is n∈N such that for all maps f1,…,fn∈F the set f1∘…∘fn(X) is contained in some set U∈U. In the paper we present some idea how to extend a topological fractal

For certain property ℜ of topological T0-spaces, we prove that if T0-space Y has property ℜ, then [X,Y] endowed with the Isbell topology possesses ℜ for any T0-space X; and if [X,Y] has property ℜ for some non-empty space X, then so does Y.

V. Bergelson and N. Hindman proved that if 〈xn〉n=1∞ is a sequence in (N,+), then for any IP⁎ set A there exists a sum subsystem 〈yn〉n=1∞ of 〈xn〉n=1∞ with the property that FS(〈yn〉n=1∞)∪FP(〈yn〉n=1∞)⊆A. While extending this combined structure, we obtain that a stronger result holds for dynamical IP⁎ sets. But for any IP⁎ set, it is unknown.

Tukey order is used to compare the cofinal complexity of partially order sets (posets). We prove that there is a 2c-sized collection of sub-posets in 2ω which forms an antichain in the sense of Tukey ordering. Using the fact that any boundedly-complete sub-poset of ωω is a Tukey quotient of ωω, we answer two open questions published in [14]. The relation between P-base and strong Pytkeev⁎ property

The present paper investigates a strong homotopy (i.e., SA-homotopy) in a finite topological adjacency category. We prove that two minimal finite spaces are SA-homotopy equivalent if and only if they are A-isomorphic in the category, and that two finite T0-spaces are SA-homotopy equivalent if and only if they have A-isomorphic cores. As an application, we prove that all simple closed KA-curves are

The function εX assigns to each point of a given continuum X the closure of the family of all continua that contain x in their interior. We define the class S(ε) of continua for which the function εX is continuous. On the other hand, we consider some natural diagram involving the function εX and commutativity of this diagram defines a class of mappings M(ε). We investigate classes S(ε) and M(ε), and

We consider, for a finite graph G, when the surjective map Confn+1(G)→Confn(G) of configuration spaces admits a section. We study when the answer depends only on the homotopy type of G, and give a complete answer. We also provide basic techniques for construction of sections.

Property A introduced by Guoliang Yu is an amenability-type property for metric spaces. In this article, we study property A for uniformly locally finite coarse spaces. Main examples of coarse spaces are a metric space, a set equipped with a discrete group action, and a sequence of finitely generated groups. The purpose of this article is to give complete proofs to related basic theorems.

We answer a question of Piotr Minc by proving that there is no compact metrizable space whose set of components contains a unique topological copy of every metrizable compactification of a ray (i.e. a half-open interval) with an arc (i.e. closed bounded interval) as the remainder. To this end we use the concept of Borel reductions coming from Invariant descriptive set theory. It follows as a corollary

We develop a new procedure for computing the twisted Reidemeister torsions of closed 3-manifolds with respect to SLn-representations. We further suggest three normalizations of the torsions and compare the normalized torsions to the preceding torsions. Moreover, we compute the twisted torsions of some Seifert 3-manifolds.

The notion of localic real functions was introduced by Gutiérrez García et al. with which pointfree forms of insertion theorems for some classes of topological spaces were obtained. In this paper, we present insertion theorems for countably paracompact frames and stratifiable frames in terms of localic real functions, yielding the insertion theorems for countably paracompact spaces and stratifiable

Banič and Kennedy (2015) [8] have drawn attention to a natural but largely unexplored field of study in the theory of inverse limits with set-valued functions, namely using bonding functions having graphs that are arcs. At the end of that paper they pose a question: If f:[0,1]→2[0,1] is an upper semi-continuous function such that G(fn) is connected for each n and G(f) is an arc, is lim←f connected

We study the Raĭkov completion Gˇ of a paratopological group G defined by Banakh-Ravsky in 2020. This permits to Banakh and Tkachenko introduce the notion of C-semicompletion, where C is a class of continuous homomorphisms of paratopological groups. We show that the product of C-semicompletions is a C-semicompletion for the product of arbitrary paratopological groups. In particular, if we consider

For a nonpseudocompact space X, the family ∑(X) of all intermediate subrings of C(X) which contain C⁎(X) contains at least 2c many distinct rings. We show that if in addition X is first countable and realcompact, then there are at least 2c many C-type intermediate rings in ∑(X) no two of which are pairwise isomorphic. With the special case X=N, it is shown that there exists a family containing c-many