We say that a Hausdorff topological space X is HČ-complete if there exists a countable family {An: n∈N} of open covers of the space X with the following property: if F is a centered family of closed subsets of X satisfying that for each n<ω there exist Fn∈F and An∈An such that Fn⊂An, then ⋂F is non-empty. We study the HČ-completeness in the Katětov extension κX of a Hausdorff space X. We determine

One of the classical results concerning differentiability of continuous functions states that the set SD of somewhere differentiable functions (i.e., functions which are differentiable at some point) is Haar-null in the space C[0,1]. By a recent result of Banakh et al., a set is Haar-null provided that there is a Borel hull B⊇A and a continuous map f:{0,1}N→C[0,1] such that f−1[B+h] is Lebesgue's null

We show that 1, N and NN are all Tukey types of countable sequential groups reducible to NN. In particular, we show that a countable sequential group X is of Tukey type NN exactly when it a nonmetrizable kℵ0-space.

In this paper a rapid non-interval P-point and a rapid non-weakly Ramsey interval P-point are constructed assuming Continuum Hypothesis or Martin's Axiom. These constructions, with the help of geometric interpretations for some needed configurations, provide the examples of rapid ultrafilters for an affirmative answer to a question of Blass, Di Nasso, and Forti.

In this paper, we consider spaces whose Higson coronae are indecomposable continua. We show that for a non-compact proper metric space X which is coarsely geodesic and has coarse bounded geometry, the Higson corona of X is an indecomposable continuum if and only if X is coarsely equivalent to the space of natural numbers. Then we give characterizations of finitely generated groups that have one or

Menger's conjecture that Menger spaces are σ-compact is false; it is true for analytic subspaces of Polish spaces and undecidable for more complex definable subspaces of Polish spaces. We define co-analytic and co-K-analytic spaces. For non-metrizable spaces, analytic Menger spaces are σ-compact, but Menger continuous images of co-analytic spaces need not be. The general co-analytic case is still open

The pinning down number pd(X) of a topological space X is the smallest cardinal κ such that for every neighborhood assignment U on X there is a set of size κ that meets every member of U. Clearly, pd(X)≤d(X) and we call X a pd-example if pd(X)

We show that a statement concerning the existence of winning strategies of limited memory in an infinite two-person topological game is equivalent to a weak version of the Singular Cardinals Hypothesis.

The shift map, denoted σ, is the self-homeomorphism of induced by the successor function n↦n+1 on ω. We prove that the isomorphism classes of σ and σ−1 cannot be separated by a Borel set in H(ω⁎), the space of all self-homeomorphisms of ω⁎ equipped with the compact-open topology. Van Douwen proved it is consistent for σ and σ−1 not to be isomorphic. Whether it is also consistent for them to be isomorphic

Let F(X) be the free topological group on a Tychonoff space X. For all natural numbers n we denote by Fn(X) the subset of F(X) consisting of all words of reduced length ≤n. In [9], the author found equivalent conditions on a metrizable space X for F3(X) to be Fréchet-Urysohn, and for Fn(X) to be Fréchet-Urysohn for n≥5. However, no equivalent condition on X for n=4 was found. Recently, we proved in

A space has σ-compact tightness if the closures of σ-compact subsets determines the topology. We consider a dense set variant that we call densely k-separable. We consider the question of whether every densely k-separable space is separable. The somewhat surprising answer is that this property, for compact spaces, implies that every dense set is separable. The path to this result relies on the known

Let X=∏α∈ω1Xα be a product of ω1 many separable metric spaces. It is proved that every C⁎-embedded subset in X is C-embedded in X under a certain set-theoretic assumption. Combining a result of E. Pol and R. Pol for N2ω, it is undecidable in ZFC that there is a (closed) C⁎-embedded but not C-embedded subset in Nω1. It is also proved in ZFC that every C-embedded subset in X is P-embedded in X. Next

We continue Gartside, Moody, and Stares' study of versions of monotone paracompactness. We show that the class of spaces with a monotone open closure-preserving operator is strictly larger than those with a monotone open locally-finite operator. We prove that monotonically metacompact GO-spaces have a monotone open star-finite operator, and so do GO-spaces with a monotone (open or not) closure-preserving

A Tychonoff space X is called strongly Čech-complete if there exist paracompact open subsets V1,V2,… of βX such that ⋂n=1∞Vn=X. Strong Čech-completeness of a Tychonoff space is characterized in terms of the existence of certain kinds of complete sequence of open covers, for example, of a complete sequence consisting of star-finite open covers. A metrizable space is shown to be strongly Čech-complete

In 1983, Mumin Dai and Shôgo Ikenaga independently introduced mutually equivalent *Lindelöf spaces and ω-star spaces, later called strong 1-star-Lindelöf spaces in 1991, star-Lindelöf spaces in 1998 or star-countable spaces in 2007. After a series of work of E.K. van Douwen, M.V. Matveev and J. van Mill et al., star covering properties, the star-compactness and star-Lindelöfness being the cores, have

A topological space X is uniquely homogeneous if for every a,b∈X, there is exactly one homeomorphism h:X→X such that h(a)=b. We say that X is strongly uniquely homogeneous if it is uniquely homogeneous and the only continuous functions f:X→X are homeomorphisms and constant functions. We show that if n≥2, then there is a strongly uniquely homogeneous subspace of Rn, all of whose homeomorphisms are restrictions

This is a survey of four problems that are “classics” in many different senses of the word, and of several related problems associated with each one. The numbering of the Classic Problems picks up where that of a similar article left off about four decades ago: IX. Is every point of ω⁎ a butterfly point? X. Is there a nonmetrizable perfectly normal, locally connected continuum? XI. Is there a normal

This paper is concerned with the Laitinen Conjecture. The conjecture predicts an answer to the Smith question [22] which reads as follows. Is it true that for a finite group G acting smoothly on a sphere with exactly two fixed points, the tangent spaces at the fixed points have always isomorphic RG-module structures defined by differentiation of the action? Using the technique of induction of group

We show that the property of having only non-weak cut points is a Whitney property, and that having only shore points is not a Whitney property. We also note that the property of having only non-cut (non-weak cut, shore) points is not a Whitney reversible property. This complements a result due to E. Matsuhashi.

Let X be a continuum. The n-fold hyperspace Cn(X), n∈N, is the family of all nonempty closed subsets of X with at most n components, topologized with the Hausdorff metric. Let μ be a strong size map for Cn(X). A strong size level is the subset μ−1(t), with t∈[0,1]. A strong size block is the subset μ−1([s,t]), with 0≤s

We define the notion of D-set in an arbitrary semigroup, and with some mild restrictions we establish its dynamical and combinatorial characterizations. Assuming a weak form of cancellation in semigroups we have shown that the Cartesian product of finitely many D-sets is a D-set. A similar partial result has been proved for Cartesian product of infinitely many D-sets. Finally, in a commutative semigroup

A well ordering ≺ of a topological space X is left-separating if {x′∈X:x′≺x} is closed in X for any x∈X. A space is left-separated if it has a left-separating well-ordering. The left-separating type ordℓ(X) of a left-separated space X is the minimum of the order types of the left-separating well-orderings of X. We prove that (1) if κ is a regular cardinal, then for each ordinal α<κ+ there is a T2 space

Let f,g:X→Y be a pair of maps of a compact, connected, oriented Riemannian manifold, with or without boundary. In this paper we prove the minimum theorem (Wecken theorem) for the epsilon Nielsen coincidence number Nε(f,g).

We prove that for each natural number m≥3, there exists a Mrówka space with a diagonal of the rank exactly m. We also construct a non-submetrizable separable Moore space with a diagonal of infinite rank.

Let γ be an open cover of a space X. If there is a metric d on X which metrizes each member of γ, then X is said to be γ-metrizable. This notion was introduced and studied by M.A. Al Shumrani in [20]. We give an example to show that there is a gap in the proof of Theorem 2.1 and we give an example of a regular γ-metrizable space which is not metrizable, showing that main Theorems 2.1 and 2.4 in Al

Hindman and Leader first introduced the notion of the semigroup of ultrafilters converging to zero for a dense subsemigroup of ((0,∞),+). Using the algebraic structure of the Stone-Čech compactification, Tootkaboni and Vahed generalized and extended this notion to an idempotent instead of zero, that is a semigroup of ultrafilters converging to an idempotent e for a dense subsemigroup of a semitopological

In [1] Borzellino and Brunsden started to develop an elementary differential topology theory for orbifolds. In this paper we carry on their project by defining a mapping degree for proper maps between orbifolds, which counts preimages of regular values with appropriate weights. We show that the mapping degree satisfies the expected invariance properties, under the assumption that the domain does not

We first introduce and study two new classes of subsets in T0 spaces — Rudin sets and WD sets lying between the class of all closures of directed subsets and that of irreducible closed subsets. Using such subsets, we define three new types of topological spaces — DC spaces, Rudin spaces and WD spaces. Rudin spaces lie between WD spaces and DC spaces, while DC spaces lie between Rudin spaces and sober

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.

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

Let X be a Tychonoff space. We show that Cp(X), the space of real-valued continuous functions on X equipped with the pointwise topology, admits a resolution (an ordered covering indexed by NN) consisting of convex compact sets that swallows the local null sequences if and only if X is countable and discrete. Then we prove (main result) (i) the weak* bidual of Cp(X) admits a resolution consisting of

Let Ps:=F2[x1,x2,…,xs] be the graded polynomial algebra over the prime field of two elements, F2, in s variables x1,x2,…,xs, each of degree 1. We are interested in the Peterson “hit” problem of finding a minimal set of generators for Ps as a graded left module over the mod-2 Steenrod algebra, A. For s⩾5, it is still open. In this paper, we study the hit problem of five variables in a generic degree

We introduce Kaestner brackets, a generalization of biquandle brackets to the case of parity biquandles. This infinite set of quantum enhancements of the biquandle counting invariant for oriented virtual knots and links includes the classical quantum invariants, the quandle and biquandle 2-cocycle invariants and the classical biquandle brackets as special cases, coinciding with them for oriented classical

Let A and B be two subspaces of an ordinal. It is proved that the product A×B is countably paracompact if and only if for every regular uncountable cardinal κ and for every closed rectangle A′×B′ in A×B, either A′ or B′ contains a closed discrete subset of size κ whenever so does A′×B′. This characterization of the countably paracompact products of ordinals is much simpler than the previous one given

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.

Let K be a nontrivial knot in S3 and t(K) its tunnel number. For any (p≥2,q)-slope in the torus boundary of a closed regular neighborhood of K in S3, denoted by K⋆, it is a nontrivial cable knot in S3. Though t(K⋆)≤t(K)+1, Example 1.1 in Section 1 shows that in some case, t(K⋆)≤t(K). So it is interesting to know when t(K⋆)=t(K)+1. After using some combinatorial techniques, we prove that (1) for any

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 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.

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

Let M be a compact surface and P be either R or S1. For a smooth map f:M→P and a closed subset V⊂M, denote by S(f,V) the group of diffeomorphisms h of M preserving f, i.e. satisfying the relation f∘h=f, and fixed on V. Let also S′(f,V) be its subgroup consisting of diffeomorphisms isotopic relatively V to the identity map idM via isotopies that are not necessarily f-preserving. The groups π0S(f,V)

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.

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

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.

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

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

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

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

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

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

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

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

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