In this article the notion of domination of metrizable spaces in the sense of diagrams has been introduced. A few of its properties and applications are given. It deserves particular attention that the new domination can be applied to the coincidence theory.

We say that a space X is dually CCC (respectively, weakly Lindelöf, separable) if for any neighbourhood assignment ϕ on X, there is a CCC (respectively, weakly Lindelöf, separable) subspace Y⊂X such that ϕ(Y)={ϕ(y):y∈Y} covers X. In this paper, we mainly show that (1) A dually CCC first countable Hausdorff space has cardinality at most 2c and a dually weakly Lindelöf first countable normal space has

We force a classification of all the Abelian groups of cardinality at most 2c that admit a countably compact group with a non-trivial convergent sequence. In particular, we answer (consistently) Question 24 of Dikranjan and Shakhmatov [10] for cardinality at most 2c, by showing that if a non-torsion Abelian group of size at most 2c admits a countably compact Hausdorff group topology, then it admits

Our main objective is a further study of M-factorizability in topological groups as defined in Zhang, Peng, He, Tkachenko (2020) [15]. We focus on topological-algebraic implications of M-factorizability such as τ-precompactness, pseudo-τ-compactness and τ-fineness. We also study products of topological groups and present necessary and sufficient conditions on the factors guaranteeing the M-factorizability

If X is a dense subspace of a space B, then B is called an extension of X, and the subspace Y = B∖X is called a remainder of X. We study below, how the properties of remainders of spaces influence the properties of these spaces. In particular, we establish the following fact: if Y is a remainder of a topological group G in an extension B of G, and every closed pseudocompact Gδ-subspace of Y is compact

We discuss degree-preserving crossing change on Khovanov homology in terms of cobordisms. Namely, using Bar-Natan's formalism of Khovanov homology, we investigate a sum of cobordisms that yields a morphism between complexes of two diagrams related by a change of crossing, which we call the “genus-one morphism.” We prove that the morphism is invariant under the moves of double points in tangle diagrams

We prove that for a positive integer m and a non-negative integer n, if the transfinite asymptotic dimension of an ∞-pseudometric space X is at most m⋅ω+n, then X has an integral APD profile (n+1,α1,…,αm) for some non-decreasing functions α1,…,αm. This answers a question of Orzechowski posed in 2020.

We show that the space of subprobability measures, equivalently of subprobability continuous valuations, on an algebraic (resp., continuous) complete quasi-metric space is again algebraic (resp., continuous) and complete, when equipped with the Kantorovich-Rubinstein quasi-metrics dKR (unbounded) or dKRa (bounded), themselves asymmetric forms of the well-known Kantorovich-Rubinstein metric. We also

In the second part of this article, we show that if Z is a metacompact subspace of a GO-space X and U is a family of open subsets of X such that Z⊂⋃U then there exists a point-finite family V of open subsets of X such that Z⊂⋃V and V≺U. Thus every subspace Y of a GO-space X is metacompact in X if and only if Y is a metacompact subspace of X. In the third part of this article, we get the following conclusions

We demonstrate a one-to-one correspondence between idempotent closure operators and the so-called saturated quasi-uniform structures on a category C. Not only this result allows to obtain a categorical counterpart P of the Császár-Pervin quasi-uniformity P, that we characterize as a transitive quasi-uniformity compatible with an idempotent interior operator, but also permits to describe those topogenous

In this paper, we investigate the existence of predecessors and successors of the usual Euclidean topology τX on X in PG(X) and G(X), where X={(ab01):a>0,a,b∈R} and G(X) (PG(X)) is the lattice of all topological (paratopological) group topologies on X. We give a complete description of predecessors of τX in G(X) based on the fact that (X,τX) is a minimal Hausdorff topological group which was shown

In this paper, we mainly consider some cardinal invariants and grasps of quasitopological groups and some properties of two classes of quasitopological groups. We show that: (1) There exists a pseudocompact quasitopological group K with countable cellularity and In(K)>ω, which gives a negative answer to [30, Question 5.3] ([27, Question 3.6]); (2) There exists a pseudocompact quasitopological group

In this paper some formulas of supremum topological sequence entropy htop∞(f) are investigated for circle maps. If f is a circle map then htop∞(f)=htop∞(f|E(f)‾), where E(f)‾ is the closure of the set of eventually periodic points of f; If f is a circle map with zero topological entropy and Fix(f)≠∅ then htop∞(f|Ω(f))=0, where Ω(f) denotes the set of non-wandering points of f.

In this paper, we answer two problems concerning sobriety. One concerns the existence of a closed set lattice whose Scott space is non-sober. Another is about the reflectivity of a category of k-bounded sober spaces. We also prove some sufficient conditions for the Scott space of a dcpo to be k-bounded well-filtered.

In this paper, we study some properties of retraction sequences, blowups of polytopes and their interrelations. We introduce the blowups of quasitoric orbifolds using the combinatorial data associated with its orbit spaces. We prove that neither new singularity nor new torsion arises in the integral cohomologies of certain blowups of a quasitoric orbifold. Moreover, we show that if a quasitoric orbifold

Let A⊂I=[0,1] such that λA=0 and A is a dense Gδ subset of [0,1], and take M to be the set of measurable self-maps of I. There exists a residual set R⊂M such that for each f in R, the following hold: (1) The range of f is contained in A, and f is a one-to-one function. (2) For any x∈[0,1], the ω-limit set ω(x,f) is nowhere dense. (3) The Hausdorff dimension of the ω-limit points Λ(f)‾=∪x∈Iω(x,f) is

The fundamental quandle is a powerful invariant of knots and links, but it is difficult to describe in detail. It is often useful to look at quotients of the quandle, especially finite quotients. One natural quotient introduced by Joyce [1] is the n-quandle. Hoste and Shanahan [2] gave a complete list of the knots and links which have finite n-quandles for some n. We introduce a generalization of n-quandles

We study the relation of continuous reducibility, or Wadge reducibility, between subsets of an affine variety. We show that on any curve the relation of continuous reducibility is a bqo, though it may have large finite antichains. We determine the Wadge hierarchy on irreducible curves and on countable irreducible affine varieties of any dimension. Under a technical assumption of adequateness, we prove

We show that the Morse boundary of a right-angled Coxeter group may contain embedded circles that do not arise as the boundary of a Morse Fuchsian subgroup visible in the defining graph.

The Gram determinant of type A was introduced by Lickorish in his work on invariants of 3 - manifolds. We generalize the theory of the Gram determinant of type A by evaluating, in the annulus, a bilinear form of non-intersecting connections in the disc. The main result provides a closed formula for this Gram determinant. We conclude the paper by discussing Chen's conjecture about the Gram determinant

Let α∈R and letA=[1101]andBα=[10α1]. The subgroup Gα of SL2(R) is a group generated by the matrices A and Bα. In this paper, we investigate the property of the group Gα. We construct a generalization of the Farey graph for the subgroup Gα. This graph determines whether the group Gα is a free group of rank 2. More precisely, the group Gα is a free group of rank 2 if and only if the graph is tree. In

This paper is the first in a two-part series. In this paper, we prove that the Assouad-Nagata dimension of any finitely generated (but not necessarily finitely presented) C′(16) group is at most 2. In the next paper, we use this result, along with techniques of classical small cancellation theory, to answer two open questions in the study of asymptotic and Assouad-Nagata dimension of finitely generated

We construct a 1-parameter family of SL2(R) representations of the pretzel knot P(−2,3,7). As a consequence, we conclude that Dehn surgeries on this knot are left-orderable for all rational surgery slopes less than 6. Furthermore, we discuss a family of knots and exhibit similar orderability results for a few other examples.

Castro Pereira and Tomita showed that it is consistent that there exist arbitrarily large countably compact groups without non-trivial convergent sequences of finite order. In this paper, we obtain large p-compact groups without non-trivial convergent on the direct sum of κ copies of Q where κ=κω is infinite. In particular this gives the first arbitrarily large countably compact groups without non-trivial

For a metric continuum X, we consider the hyperspace of subcontinua C(X) of X, with the Hausdorff metric. A Whitney mapping is a continuous function μ:C(X)→[0,∞) such that: (a) for each p∈X, μ(p)=0, and (b) if A,B∈C(X) and A⊊B, then μ(A)<μ(B). The Whitney mapping μ is finitely generated if there exist a finite number of continuous functions f1,…,fn:X→[0,1] such that for each A∈C(X), μ(A)=length(f1

We prove that basket links, whose symmetrized Seifert form is congruent to the Cartan matrix of the simply laced Dynkin diagram An, are isotopic to the torus link T(2,n+1). In addition, we provide examples of links, constructed by plumbing n positive Hopf bands the core curves of which intersect at most once, with symmetrized Seifert form congruent to the Cartan matrix An, that are not isotopic to

Given a compact metric space X and a continuous function f:X→X, we introduce the set function Tf:X→2X given by Tf(x)=T({f(x)}), where 2X is the family of nonempty closed subsets of X and T is Jones' set function. We consider the dynamical system (X,Tf) and study its properties.

We introduce average shadowable measures and almost average shadowable measures for continuous maps of compact metric spaces and weakly topologically stable points and average persistent property for homeomorphisms of compact metric spaces. We prove that the set of all average shadowable measures is dense in the space of all Borel probability measures if and only if the set of all average shadowable

In this paper, we study the relationship between complementary-finite asymptotic dimension and transfinite asymptotic dimension. We prove that for every metric space X, coasdim(X)≤γ implies trasdim(X)≤γ for every countable ordinal number γ. Therefore, the metric space with complementary-finite asymptotic dimension has asymptotic property C. Furthermore, we construct an example of a metric space satisfying

We characterize the class of uniform limits of functions from Pawlak's class B1⁎⁎. The resulting class uS1, which contains functions with the oscillation rank one, is discussed in connection with its linear span. We apply a general, topological space, setting in our discussion.

We give lower bounds for the tunnel number of knots and handlebody-knots. We also give a lower bound for the cutting number, which is a “dual” notion to the tunnel number in handlebody-knot theory. We provide necessary conditions to be constituent handlebody-knots by using G-family of quandles colorings. The above two evaluations are obtained as the corollaries. Furthermore, we construct handlebody-knots

Given a discrete group G, we identify the Stone-Čech compactification βG with the set of all ultrafilters on G and put G⁎=βG∖G. The action G on G by the conjugations (g,x)↦g−1xg induces the action of G on G⁎ by (g,p)↦pg, pg={g−1Pg:P∈p}. We study interplays between the algebraic properties of G and the dynamical properties of (G,G⁎). In particular, we show that pG is finite for each p∈G⁎ if and only

We discuss the class of paratopological groups which admits a transversal, T1-independent and T1-complementary paratopological group topology. We show that the Sorgenfrey line does not admit a T1-complementary Hausdorff paratopological group topology, which gives a negative answer to [5, Problem 10]. We give a very useful criterion for transversality in term of submaximal paratopological group topology

It is well known in [3] that the liminf convergence is topological in a domain. We supply an example to illustrate that a dcpo in which the liminf convergence is topological may not be continuous in this paper. Naturally, there is a problem raised as follows: in what posets is the liminf convergence topological? First, we show that if the liminf convergence is topological in a poset, then the poset

Motivated by Akbulut-Larson's construction of Brieskorn spheres bounding rational homology 4-balls, we explore plumbed 3-manifolds that bound rational homology circles and use them to construct infinite families of rational homology 3-spheres that bound rational homology 4-balls. Some of these rational homology 3-spheres are new examples of integer homology 3-spheres that bound rational homology 4-balls

In this paper, we study metrizable and weakly metrizable coset spaces. It is mainly shown that (1) If H is a closed neutral subgroup of a topological group G, then G/H is metrizable ⇔ G/H is bisequential ⇔ G/H is weakly first-countable ⇔ G/H is a Fréchet-Urysohn space with an ωω-base; (2) If H is a closed neutral subgroup of a semitopological group G, then G/H is metrizable if and only if G/H is a

We employ the relative category to develop relations between the Ważewski pair (N,E) and the Morse decomposition of the maximal invariant set in N∖E‾ for infinite-dimensional dynamical systems. Via these relations, we can detect connecting trajectories between Morse sets and obtain a dynamical-system version of critical point theorem with relative category.

In this paper, we prove the following results: (1) Let f:G→H be a continuous d-open surjective homomorphism; if G is an R-factorizable paratopological group, then so is H. This improves a result by Peng and Zhang [4, Theorem 1.7], and (2) let G be a regular (so G is completely regular by a theorem of Banakh and Ravsky.) R-factorizable paratopological group; then every subgroup H of G is R-factorizable

The Kestelman-Borwein-Ditor Theorem asserts that a non-negligible subset of R which is Baire (= has the Baire property, BP) or measurable is shift-compact: it contains some subsequence of any null sequence to within translation by an element of the set. Here effective proofs are recognized to yield (i) analogous category and Haar-measure metrizable generalizations for Baire groups and locally compact

The notion of D-sublocale is explored. This is the notion analogue to that of sublocale in the duality of TD-spaces. A sublocale S of a frame L is a D-sublocale if and only if the corresponding localic map preserves the property of being a covered prime. It is shown that for a frame L the system of those sublocales which are also D-sublocales form a dense sublocale SD(L) of the coframe S(L) of all

In this paper we study topological properties of Lyapunov stable sets for semigroup actions on Tychonoff spaces. We show that the stability and the asymptotical stability of a compact set is characterized by the stability and the asymptotical stability of its components, respectively. We also present a characterization of the stability of a compact set by means of ω-limit sets, prolongations, prolongational

We study a combinatorial homotopy theory for small categories without a loop (loopfree categories) which is closely related to Whitehead's simple homotopy theory for regular CW-complexes with triangular cells. Quillen's theorem A and the nerve theorem for loopfree categories are considered from the viewpoint of the simple homotopy theory. Moreover, we extend the classical nerve theorem and discuss

Motivated by constructions in topological data analysis and algebraic combinatorics, we study homotopy theory on the category of Čech closure spaces Cl, the category whose objects are sets endowed with a Čech closure operator and whose morphisms are the continuous maps between them. We introduce new classes of Čech closure structures on metric spaces, graphs, and simplicial complexes, and we show how

Let p denote an odd prime. We show by example that the inequalities obtained in [6] for the behaviour of the cohomological index of a join of free Z/p-actions are sharp. Namely, for all odd integers k,l at least one of which is greater than one, we give examples of finite free Z/p-CW complexes of cohomological indices k and l whose join has index k+l+1 and also examples where the join has index k+l−1

Let D(M,N) be the set of degrees of maps from M to N. In this article, we determine D(M,N) when M and N are S3-bundles over S5. It also provides a correction of an article by Lafont and Neofytidis.

Let C(X) be the hyperspace of all subcontinua of a metric continuum X. An element A∈C(X) makes a hole in C(X) if C(X)−{A} is not unicoherent. In this paper, we characterize the elements A∈C(X) satisfying that A makes a hole in C(X) when X has property (b).

Vector fields and line fields, their counterparts without orientations on tangent lines, are familiar objects in the theory of dynamical systems. Among the techniques used in their study, the Morse–Smale decomposition of a (generic) field plays a fundamental role, relating the geometric structure of phase space to a combinatorial object consisting of critical points and separatrices. Such concepts

We raise the problem of realisability of rings as {X,X} the ring of stable homotopy classes of self-maps of a space X. By focusing on An2-polyhedra, we show that the direct sum of three endomorphism rings of abelian groups, one of which must be free, is realisable as {X,X} modulo the acyclic maps. We also show that Fp3 is not realisable in the setting of finite type An2-polyhedra, for p any prime.

An algebraically fibering group is an algebraic generalization of the fibered 3-manifold group in higher dimensions. Let M(P) and M(E) be the cusped and compact hyperbolic real moment-angled manifolds associated with the hyperbolic right-angled 24-cell P and the hyperbolic right-angled 120-cell E, respectively. Jankiewicz, Norin, and Wise recently showed that π1(M(P)) and π1(M(E)) are algebraically

We introduce new formulas that are Vassiliev invariants of flat vertex isotopy classes of 2-bouquet graphs. Although any Gauss diagram formula of Vassiliev invariants of 2-bouquet graphs in a 3-space has been unknown explicitly, this paper gives the first and simple example.

This paper investigates a constructive characterization of kneading sequences belonging to unimodal maps with embedded adding machines. It takes the concept of a {0,1}-regular scheme, which can be used to generate all kneading sequences of unimodal maps for which the turning point is regularly recurrent, and adds an extra condition that guarantees the constructed kneading sequence will additionally

Let Gk(PSp(2)) and Gk(PSp(3)) be the gauge groups of principal PSp(2)-bundles over S8 and principal PSp(3)-bundles over S4 classified by kε1′ and kε2′, where ε1′ and ε2′ are generators of π8(BPSp(2))≅Z and π4(BPSp(3))≅Z, respectively. In this article we partially classify the homotopy types for these gauge groups.

The main purpose of this note is to prove that the product of a cellular-Lindelöf space with a space of countable spread need not be cellular-Lindelöf.

We calculate the asymptotic behavior of the Kashaev invariant of a twice-itarated torus knot and obtain topological interpretation of the formula in terms of the Chern–Simons invariant and the twisted Reidemeister torsion.

In this paper we extend the notion of a clopen subset of a topological space to that of a clopen subobject in a topological category. We relate clopen subobjects to the concept of normalized topological category via a theorem which, in turn, shows why certain subsets must be clopen in a topological space. We employ clopen subobjects to extend the topological notion of connectedness to topological categories

We prove that there exists a model of ZF (Zermelo–Fraenkel set theory without the Axiom of Choice (AC)) in which there is a compact, metrizable, non-second countable, Cantor cube. This answers in the affirmative an open question by E. Wajch (2018) [15]. Furthermore, we strengthen a result in the above paper, namely “Countable products of metrizable spaces are quasi-metrizable implies van Douwen's Choice

2-stratifolds are a generalization of 2-manifolds that occur as objects in applications such as in TDA. These spaces can be described by an associated bicoloured labelled graph. In previous papers we obtained a classification of 1-connected trivalent 2-stratifolds. In this paper we classify trivalent 2-stratifolds that have fundamental group Z. This classification implicitly gives an efficient algorithm

A simple closed curve α in the boundary of a genus two handlebody H is primitive if adding a 2-handle to H along α yields a solid torus. If adding a 2-handle to H along α yields a Seifert-fibered space and not a solid torus, the curve is called Seifert. If α is disjoint from an essential separating disk in H, does not bound a disk in H, and is not primitive in H, then it is said to be proper power

Let Pk:=F2[x1,x2,…,xk] be the polynomial algebra over the prime field of two elements, F2, in k variables x1,x2,…,xk, each of degree 1. We study the hit problem, set up by Frank Peterson, of finding a minimal set of generators for Pk as a module over the mod-2 Steenrod algebra. In this paper, we extend a result in [12] on the hit problem in degree (k−1)(2d−1) with k⩾6, by explicitly computing the hit

We study expansive dynamical systems in the setting of distributive lattices and their automorphisms, the usual notion of expansiveness for a homeomorphism of a compact metric space being the particular case when the lattice is the topology of the phase space ordered by inclusion and the automorphism the one induced by the homeomorphism, mapping open sets to open sets. We prove in this context generalizations