Given a compact metric space X and a continuous function f:X→X, we define the ω-limit function ωf:X→2X by ωf(x)=ω(x,f) for each x∈X. We study the continuity of ωf when f is defined on a dendrite. Furthermore, we show some relationships between the connectedness of the set of periodic points and the equicontinuity of the function.

The purpose of this paper is to develop a theory of \((\infty , 1)\)-stacks, in the sense of Hirschowitz–Simpson’s ‘Descent Pour Les n–Champs’, using the language of quasi-category theory and the author’s local Joyal model structure. The main result is a characterization of \((\infty , 1)\)-stacks in terms of mapping space presheaves. An important special case of this theorem gives a sufficient condition

We use a Borsuk-Ulam type argument in order to prove existence of nontrivial bounded solutions to some nonautonomous differential equations, which are odd with respect to the spatial variable.

We show that every alternating pretzel knot satisfies the slice-ribbon conjecture and that every alternating pretzel link with m components (m≥2) is not slice in the strong sense.

Let D be a dendrite with unique branch point and f:D→D be continuous. Denote by R(f) and Ω(f) the set of recurrent points and the set of non-wandering points of f, respectively. Let Ω0(f)=D and Ωk(f)=Ω(f|Ωk−1(f)) for any positive integer k. The minimal k such that Ωk(f)=Ωk+1(f) is called the depth of f, where k is a positive integer or ∞. In this note, we show that Ω2(f)=R(f)‾ and the depth of f is

We consider the problem of realizing abstract graph symmetries by symmetries of the 3-sphere through spatial embeddings. The techniques usually used in preceding studies are based on the placement of vertices and edges, and the problem has been solved for several well-known families of graphs. This paper provides a technique which is applicable to any graph, and gives an affirmative answer to the problem

We describe the topological structure of closed manifolds of dimension no less than four which admit Morse-Smale diffeomorphisms such that its non-wandering set contains any number of sink periodic points, and any number of source periodic points, and few saddle periodic points.

A nonempty compact saturated subset F of a topological space is called k-irreducible if it cannot be written as a union of two compact saturated proper subsets. A topological space is said to be co-sober if each of its k-irreducible compact saturated sets is the saturation of a point. Wen and Xu (2018) proved that Isbell's non-sober complete lattice equipped with the lower topology is sober but not

We introduce the notion of directed diagrammatic reducibility which is a relative version of diagrammatic reducibility. Directed diagrammatic reducibility has strong group theoretic and topological consequences. A multi-relator version of the Freiheitssatz in the presence of directed diagrammatic reducibility is given. Results concerning asphericity and π1-injectivity of subcomplexes are shown. We

We show that for any map f on an arc-like continuum X, the induced map fˆ on the hyperspace of subcontinua C(X) fixes a point in any fˆ-invariant subcontinuum of C(X). This extends a result of Robatian [21], who proved it for the arc. However, as we show, the result does not extend to tree-like continua. We conclude with a list of related problems. Our proof builds on Hamilton's proof of the fixed

We consider generalized inverse limits of continua with bonding functions Fn that have the projection of Graph(Fn) onto the second (first) factor atomic and images (pre-image) of points are zero-dimensional. For such bonding functions we show that under some easily verified conditions that if the first (all) factor space(s) has a certain property then the inverse limit space must have this property

This paper investigates the structure of the kneading sequences of unimodal maps for which the omega-limit set of the turning point is a Cantor set and the map restricted to that omega-limit set is a minimal homeomorphism. We provide several characterizations of unimodal maps with these homeomorphic restrictions in terms of the kneading sequences and the associated shift spaces generated by the kneading

For a virtual n-link K, we define a new virtual link VD(K), which is invariant under virtual equivalence of K. The Dehn space of VD(K), which we denote by DD(K), therefore has a homotopy type which is an invariant of K. We show that the quandle and even the fundamental group of this space are able to detect the virtual trefoil. We also consider applications to higher-dimensional virtual links.

Given an infinite cardinal κ, we prove that a regular space X must have density not exceeding κ if and only if every subset of X of power (2κ)+ is κ-dominated, i.e., contained in the closure of a set of cardinality κ. Thus, it is natural to study the spaces in which all subsets of cardinality not exceeding 2κ are κ-dominated; this property will be called exponential κ-domination. The spaces with exponential

We characterize the left-handed noncommutative frames that arise from sheaves on topological spaces. Further, we show that a general left-handed noncommutative frame A arises from a sheaf on the dissolution locale associated to the commutative shadow of A. Both constructions are made precise in terms of dual equivalences of categories, similar to the duality result for strongly distributive skew lattices

Lambda-\({\mathcal {S}}\) is an extension to first-order lambda calculus unifying two approaches of non-cloning in quantum lambda-calculi. One is to forbid duplication of variables, while the other is to consider all lambda-terms as algebraic linear functions. The type system of Lambda-\({\mathcal {S}}\) has a constructor S such that a type A is considered as the base of a vector space while S(A) is

We study track categories (i.e., groupoid-enriched categories) endowed with additive structure similar to that of a 1-truncated DG-category, except that composition is not assumed right linear. We show that if such a track category is right linear up to suitably coherent correction tracks, then it is weakly equivalent to a 1-truncated DG-category. This generalizes work of the first author on the strictification

In 1985 M. Smith constructed a nonmetric pseudo-arc; i.e. a Hausdorff homogeneous, hereditary equivalent and hereditary indecomposable continuum. Taking advantage of a decomposition theorem of W. Lewis, he obtained it as a long inverse limit of metric pseudo-arcs with monotone bonding maps. Extending his approach, and the results of Lewis on lifting homeomorphisms, we construct a nonmetric pseudo-circle

The Burau representation is a fundamental bridge between the braid group and diverse other topics in mathematics. A 1974 question of Birman asks for a description of the image; in this paper we give an approximate answer. Since a 1984 paper of Squier it has been known that the Burau representation preserves a certain Hermitian form. We show that the Burau image is dense in this unitary group relative

This paper introduces the notion of a ‘pseudo centre’ of a suitable bounded region in Rn. We investigate general properties of a pseudo centre by means of techniques provided by calculus, differential geometry, and topology. While calculus allows to introduce ‘depth diagram’ and relates that to constraint extreme value problems, the topological point of view allows to study the qualitative properties

In this paper we introduce concepts of higher equivariant and invariant topological complexities and study their properties. Then we compare them with equivariant LS-category. We give lower and upper bounds for these new invariants. We compute some of these invariants for moment angle complexes.

We provide criteria ensuring that a tunnel number one knot K is not determined by its double branched cover, in the sense that the double branched cover is also the double branched cover of a knot \(K'\) not equivalent to K.

We study linear series on a general curve of genus g, whose images are exceptional with respect to their secant planes. Each such exceptional secant plane is algebraically encoded by an included linear series, whose number of base points computes the incidence degree of the corresponding secant plane. With enumerative applications in mind, we construct a moduli scheme of inclusions of limit linear

Dilation surfaces are generalizations of translation surfaces where the geometric structure is modelled on the complex plane up to affine maps whose linear part is real. They are the geometric framework to study suspensions of affine interval exchange maps. However, though the \(SL(2,\mathbb {R})\)-action is ergodic in connected components of strata of translation surfaces, there may be mutually disjoint

Let X be a Hausdorff topological space and let Q(X,R) be the space of all quasicontinuous functions on X with values in R and τp be the topology of pointwise convergence. We prove that Q(X,R) is dense in RX equipped with the product topology. We characterize some cardinal invariants of (Q(X,R),τp). We also compare cardinal invariants of (Q(R,R),τp) and (C(R,R),τp), the space of all continuous functions

We describe the topological Hochschild homology of the periodic complex K-theory spectrum, THH(KU), as a commutative KU-algebra: it is equivalent to KU[K(Z,3)] and to F(ΣKUQ), where F is the free commutative KU-algebra functor on a KU-module. Moreover, F(ΣKUQ)≃KU∨ΣKUQ, a square-zero extension. In order to prove these results, we first establish that topological Hochschild homology commutes, as an algebra

The main aim of this article is to develop a categorical duality between the category of semilattices with homomorphisms and a category of certain topological spaces with certain morphisms. The principal tool to achieve this goal is the notion of irreducible filter. Then, we apply this dual equivalence to obtain a topological duality for the category of bounded lattices and lattice homomorphism. We

We investigate the so-called order-sobrification monad proposed by Ho et al. (Log Methods Comput Sci 14:1–19, 2018) for solving the Ho–Zhao problem, and show that this monad is commutative. We also show that the Eilenberg–Moore algebras of the order-sobrification monad over dcpo’s are precisely the strongly complete dcpo’s and the algebra homomorphisms are those Scott-continuous functions preserving

Let X=(X,d,M,μ) be a metric measure space, where d is a metric on X, M is a σ-algebra of X, and μ is a measure on M. Suppose that X satisfies the following conditions: (1) X is separable and locally compact; (2) M contains the Borel sets of X and for each E∈M, there exists a Borel set B⊂X such that E⊂B and μ(B∖E)=0; (3) for every non-empty open set U⊂X, μ(U)>0; (4) for all compact sets K⊂X, μ(K)<∞;

In the paper a general and compact fixed point theorem for dislocated (or partial) metric spaces is proved in a simple way. It appears that the reasoning is sufficiently general to spread over the well known (and complicated) classical results. In consequence, the extensions of some celebrated fixed point theorems are obtained, while their proofs are short and consistent, which is the main aim of the

We study three different topologies on the moduli space \(\mathcal {H}^\mathrm{loc}_m\) of equivariant local isometry classes of m-dimensional locally homogeneous Riemannian spaces. As an application, we provide the first examples of locally homogeneous spaces converging to a limit space in the pointed \(\mathcal {C}^{k,\alpha }\)-topology, for some \(k>1\), which do not admit any convergent subsequence

In this paper, several equivalent conditions are given for any c-expansive map with the shadowing property to be degenerate, including that the derived degree of the space is an integer. We also give a topological characterization of a countable compactum which admits an expansive homeomorphism with the shadowing property.

Given any quasi-metric space X,d, we can form the space LX of all lower semicontinuous maps from X to R‾+, where X is given the d-Scott topology. We give LX the Scott topology of the pointwise ordering. We can then form the subspace Lα(X,d) of α-Lipschitz continuous maps from X,d to R‾+ (α∈R+). We show that, when X,d is continuous Yoneda-complete, Lα(X,d) is stably compact, and that its topology coincides

In this article we obtain the rigid isotopy classification of generic rational curves of degre 5 in \({\mathbb {R}}{\mathbb {P}}^{2}\). In order to study the rigid isotopy classes of nodal rational curves of degree 5 in \({\mathbb {R}}{\mathbb {P}}^{2}\), we associate to every real rational nodal quintic curve with a marked real nodal point a nodal trigonal curve in the Hirzebruch surface \(\Sigma

Given a continuum X and p∈X, we consider the hyperspace HS(p,X) defined as the quotient space C(X)/C(p,X), where C(X) is the hyperspace of subcontinua of X and C(p,X) is the subspace of all elements in C(X) containing p. For a mapping f:X→Y between continua, let C(f):C(X)→C(Y) given by C(f)(A)=f(A), this mapping induces a natural function HS(p,f):HS(p,X)→HS(f(p),Y). In this paper we present all the

A topological space is called Loeb if the collection of all its non-empty closed sets has a choice function. Loeb and sequential spaces are investigated in the absence of the axiom of choice. Among other results, the following theorems are proved in ZF: (i) if X is a Cantor completely metrizable second-countable space, then Xω is Loeb; (ii) if a sequential, locally sequentially compact space X has

Suppose that for each i≥0, Ii=[0,1] and f is an inverse sequence of upper semicontinuous surjective functions fi+1:Ii+1→2Ii, each with a path-connected graph. We investigate necessary and sufficient conditions for lim←f to be path-connected. We define the notion of a path-component base and show that if f admits a path-component base then lim←f is not path-connected, and we give a characterisation

We study smooth rational closed embeddings of the real affine line into the real affine plane, that is algebraic rational maps from the real affine line to the real affine plane which induce smooth closed embeddings of the real euclidean line into the real euclidean plane. We consider these up to equivalence under the group of birational automorphisms of the real affine plane which are diffeomorphisms

Let κ be an infinite cardinal. A topological space X is κ-bounded if the closure of any subset of cardinality ≤κ in X is compact. We discuss the problem of embeddability of topological spaces into Hausdorff (Urysohn, regular) κ-bounded spaces, and present a canonical construction of such an embedding. Also we construct a (consistent) example of a sequentially compact separable regular space that cannot

In this note we prove that the set-Menger, set-Hurewicz and set-Rothberger properties, recently introduced in [1], are actually another view of the Menger, Hurewicz and Rothberger properties, while the weakly set-Menger and weakly set-Rothberger properties are another view of quasi-Menger and quasi-Rothberger properties [2], respectively. We also prove that the almost set-Menger property in [1] is

Let X be a smooth complex projective curve, and let \(x\,\in \, X\) be a point. We compute the automorphism group of the moduli space of framed vector bundles on X of rank \(r\, \ge \, 2\) with a framing over x. It is shown that this automorphism group is generated by the following three: (1) pullbacks using automorphisms of the curve X that fix the marked point x, (2) tensorization with certain line

We investigate the class of geodesic metric discs satisfying a uniform quadratic isoperimetric inequality and uniform bounds on the length of the boundary circle. We show that the closure of this class as a subset of Gromov-Hausdorff space is intimately related to the class of geodesic metric disc retracts satisfying comparable bounds. This kind of discs naturally come up in the context of the solution

Let A be a finite abelian p-group of rank at least 2. We show that \(E^0(BA)/I_{tr}\), the quotient of the Morava E-cohomology of A by the ideal generated by the image of the transfers along all proper subgroups, contains p-torsion. The proof makes use of transchromatic character theory.

We give sharp upper bounds on the injectivity radii of complete hyperbolic surfaces of finite area with some geodesic boundary components. The given bounds are over all such surfaces with any fixed topology; in particular, boundary lengths are not fixed. This extends the first author’s earlier result to the with-boundary setting. In the second part of the paper we comment on another direction for extending

In this note we show that if G is a countably infinite abelian group such that nG=0 for some integer n, then the only locally minimal group topology on G is the discrete one. This answers a question posed by D. Dikranjan and M. Megrelishvili in [7], in particular, a question posed by L. Außenhofer, M. Jesús Chasco, D. Dikranjan and X. Domínguez in [1]. Moreover, we give a necessary and sufficient condition

We prove that any Fano manifold of coindex three admitting nef tangent bundle is homogeneous.

When quantifying the topological properties of a metric dataset using persistent homology, the first step is to produce a simplicial filtration on the data. The Čech and Vietoris-Rips filtrations are two of the workhorses of persistent homology, but over time, various other filtrations which capture different properties of data or have different computational burdens have been introduced. Towards a

Topological properties of the free topological group and the free abelian topological group on a space have been thoroughly studied since the 1940s. In this paper, we study the free topological R-vector space V(X) on X. We show that V(X) is a quotient of the free abelian topological group on [−1,1]×X, and use this to prove topological vector space analogues of existing results for free topological

Given a simple closed curve \(\gamma \) on a connected, oriented, closed surface S of negative Euler characteristic, Mirzakhani showed that the set of points in the moduli space of hyperbolic structures on S having a simple closed geodesic of length L of the same topological type as \(\gamma \) equidistributes with respect to a natural probability measure as \(L \rightarrow \infty \). We prove several

Immersed surface-links can be described by ch-graphs derived from marked graph diagrams, analogously to the case of surface-links. By using the ch-graph description, we enumerate immersed surface-links in 4-space, up to ch-index 9.

Gelfand-Naimark-Stone duality provides an algebraic counterpart of compact Hausdorff spaces in the form of uniformly complete bounded archimedean ℓ-algebras. In [5] we extended this duality to completely regular spaces. In this article we use this extension to characterize normal, Lindelöf, and locally compact Hausdorff spaces. Our approach gives a different perspective on the classical theorems of

We first introduce and study two new classes of subsets in T0 spaces — ω-Rudin sets and ω-well-filtered determined sets lying between the class of all closures of countable directed subsets and that of irreducible closed subsets, and two new types of spaces — ω-d-spaces and ω-well-filtered spaces. We prove that an ω-well-filtered T0 space is locally compact iff it is core compact. One immediate corollary

In this paper, we introduce a new type of F-equicontinuity, which is the opposite of the existence of kF-sensitive pairs almost everywhere and show an analogue of Auslander-Yorke dichotomy theorem. Precisely, under the condition that kF is a translation invariant family, we prove that a transitive system either has F-sensitive pairs almost everywhere or is almost kF-equicontinuous of new type. Also

In this paper we study the selection principle of closed discrete selection, first researched by Tkachuk in [12] and strengthened by Clontz, Holshouser in [3], in set-open topologies on the space of continuous real-valued functions. Adapting the techniques involving point-picking games on X and Cp(X), the current authors showed similar equivalences in [1] involving the compact subsets of X and Ck(X)

Under Martin's Axiom we construct a Boolean countably compact topological group whose square is not countably pracompact.

The reduced ring order (rr-order) is a natural partial order on a reduced ring R given by r≤rrs if r2=rs. It can be studied algebraically or topologically in rings of the form C(X). The focus here is on those reduced rings in which each pair of elements has an infimum in the rr-order, and what this implies for X. A space X is called rr-good if C(X) has this property. Surprisingly both locally connected

Let Y be a metrizable space containing at least two points, and let X be a YI-Tychonoff space for some ideal I of compact sets of X. Denote by CI(X,Y) the space of continuous functions from X to Y endowed with the I-open topology. We prove that CI(X,Y) is Fréchet–Urysohn iff X has the property γI. We characterize zero-dimensional Tychonoff spaces X for which the space CI(X,2) is sequential. Extending

H.J.K. Junnila [9] called a neighbournet N on a topological space X unsymmetric provided that for each x,y∈X with y∈(N∩N−1)(x) we have that N(x)=N(y). Motivated by this definition, we shall call a T0-quasi-metric d on a set X unsymmetric provided that for each x,y,z∈X the following variant of the triangle inequality holds: d(x,z)≤d(x,y)∨d(y,x)∨d(y,z). Each T0-ultra-quasi-metric is unsymmetric. We also

Recently M.M. Choban et al. introduced spaces with fragmentable open sets and such spaces are denoted by fos-spaces. In this paper the properties of fos-spaces are studied. Among other things we show that a space covered by a locally countable family of closed fos-subspaces is a fos-space. We also prove that every Čech complete fos-space contains a dense completely metrizable space, which improves

Let X be a completely regular topological space. We study closed ideals H of CB(X), the normed algebra of bounded continuous scalar-valued mappings on X equipped with pointwise addition and multiplication and the supremum norm, which are non-vanishing, in the sense that, there is no point of X at which every element of H vanishes. This is done by studying the (unique) locally compact Hausdorff space