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This article has been retracted: please see Elsevier Policy on Article Withdrawal (https://www.elsevier.com/about/policies/article-withdrawal). This article has been retracted at the request of the Editors-in-Chief after receiving a complaint about citation issues.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PhD thesis of Z. Furdzik [O własnościach pewnych rozkładów przestrzeni topologicznych na iloczyny kartezjańskie, 1968 (unpublished, in Polish), Institute of Mathematics of the Polish Academy of Sciences, Warsaw], without proper citation.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]

This article has been retracted: please see Elsevier Policy on Article Withdrawal (https://www.elsevier.com/about/policies/article-withdrawal). This article has been retracted at the request of the Editors-in-Chief after receiving a complaint about citation issues.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PhD thesis of Z. Furdzik [O własnościach pewnych rozkładów przestrzeni topologicznych na iloczyny kartezjańskie, 1968 (unpublished, in Polish), Institute of Mathematics of the Polish Academy of Sciences, Warsaw], without proper citation.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 construct consistent examples of subgroups of 2 ω with Menger remainders which fail to have other stronger combinatorial covering properties. This answers several open questions asked by Bella, Tokgoz and Zdomskyy (Arch. Math. Logic 55 (2016), 767–784). [ABSTRACT FROM AUTHOR]

Abstract In 1995 Heath, Keppelmann and Wong studied a Nielsen type number N F (f , p) for a fibre preserving map f of fibration p. This number is a lower bound for the least number of fixed points within the fibre homotopy class of f. In this paper we generalize these ideas from fixed point theory to periodic point theory, and define two Nielsen type numbers N P n F (f , p) and N Φ n F (f , p) for periodic points of a fibre preserving map. These numbers, which can be thought of as the dual of periodic Nielsen type numbers for a relative map due to Heath, Schirmer and You, are bigger than the ordinary Nielsen type numbers. The definition of N P n F (f , p) and N Φ n F (f , p) is reminiscent of the naive type addition formulae for periodic points of fibre preserving maps due to Heath and Keppelmann. Some calculations and examples are given. [ABSTRACT FROM AUTHOR]

Abstract In this paper we introduce a notion of sensitivity for topological dynamical systems and show some of its basic features and relation to dynamical properties such as transitivity and weak mixing. Finally, we will restrict our attention to the related class of weakly positively expansive dynamical systems. [ABSTRACT FROM AUTHOR]

The multiplication of a semitopological (quasitopological) group G is called sequentially continuous if the product map of G × G into G is sequentially continuous. In this paper, we mainly consider the properties of semitopological (quasitopological) groups with sequentially continuous multiplications and three-space problems in quasitopological groups. It is showed that (1) every snf -countable semitopological group G with the sequentially continuous multiplication is sof -countable; (2) if G is a sequential quasitopological group with the sequentially continuous multiplication, then G contains a closed copy of S ω if and only if it contains a closed copy of S 2 , which give a partial answer to a problem posed by R.-X. Shen; (3) let G be a quasitopological group with the sequentially continuous multiplication, then the following are equivalent: (i) G is a sequential α 4 -space; (ii) G is Fréchet; (iii) G is strongly Fréchet; (4) (MA+¬CH) there exists a non-metrizable, separable, normal and Moore quasitopological group; (5) some examples are constructed to show that metrizability, first-countability and second-countability are not three-space properties in the class of quasitopological groups. [ABSTRACT FROM AUTHOR]

In recent work, Hess and Shipley [18] defined a theory of topological coHochschild homology (coTHH) for coalgebras. In this paper we develop computational tools to study this new theory. In particular, we prove a Hochschild–Kostant–Rosenberg type theorem in the cofree case for differential graded coalgebras. We also develop a coBökstedt spectral sequence to compute the homology of coTHH for coalgebra spectra. We use a coalgebra structure on this spectral sequence to produce several computations. [ABSTRACT FROM AUTHOR]

Given a metric continuum X , we define the hyperspace of regular subcontinua of X as the collection of all regular closed subcontinua of X . This is a new hyperspace for X . In this paper we study the connectedness, compactness, arcwise connectedness and contractibility of this hyperspace. Also we raise some open questions. [ABSTRACT FROM AUTHOR]

Let X be a CW complex, E ( X ) the group of homotopy classes of homotopy self-equivalences of X and Ψ : E ( X ) → aut ( H ⁎ ( X , Z ) ) the map sending [ α ] to H ⁎ ( α ) . This paper deals with the following question: Characterize f ⁎ ∈ aut ( H ⁎ ( X , Z ) ) such that f ⁎ ∈ Im Ψ . For the R -localized X R of an ( n + 1 ) -connected and ( 3 n + 2 ) -dimensional CW-complex X ; n ≥ 2 , where R is a certain subring of Q we define the notion of strong automorphism in aut ( H ⁎ ( X , Z ) ) , in term of the Whitehead exact sequence of the Anick model of X R and we show that f ⁎ ∈ Im Φ if and only if f ⁎ is a strong automorphism. Consequently we prove that E ( X R ) ( E ⁎ ( X ) ) R ≅ B ( X R ) , where E ⁎ ( X ) is the subgroup of the elements that induce the identity on homology and B ( X R ) is the subgroup of the strong automorphisms. [ABSTRACT FROM AUTHOR]

In this paper, we consider a common generalization of both hypercontinuous posets and s 2 -quasicontinuous posets, and introduce a new concept of quasi-hypercontinuous posets. The main results are: (1) A poset is a quasi-hypercontinuous poset iff the upper topology is a hypercontinuous lattice iff the S ⁎ -convergence is topological with respect to the upper topology; (2) For a quasi-hypercontinuous poset, the quasi-liminf convergence is topological with respect to the interval topology; (3) The interval topology on a quasi-hypercontinuous poset is Tychonoff; (4) A lattice is quasi-hypercontinuous as a poset iff its normal completion is a quasi-hypercontinuous lattice. [ABSTRACT FROM AUTHOR]

A semitopological group (topological group) is a group endowed with a topology for which multiplication is separately continuous (multiplication is jointly continuous and inversion is continuous). In this paper we show that many Baire semitopological groups are in fact topological groups. [ABSTRACT FROM AUTHOR]

A filter F on ω is called Fréchet–Urysohn if the space with only one non-isolated point ω ∪ { F } is a Fréchet–Urysohn space, where the neighborhoods of the non-isolated point are determined by the elements of F . In this paper, we distinguish some Fréchet–Urysohn filters by using two pre-orderings of filters: One is the Rudin–Keisler pre-order and the other one was introduced by Todorčević–Uzcátegui in [11] . We mainly construct an RK -chain of size c + which is RK -above of every FU -filter. Also, we show that there is an infinite RK -antichain of FU -filters. [ABSTRACT FROM AUTHOR]

In this paper we study the hyperspace of some kinds of non-cut sets of continua. This complements the study that has been done for non-cut points in continua by several authors. [ABSTRACT FROM AUTHOR]

A theorem of Hoischen states that given a positive continuous function ε : R n → R , an unbounded sequence 0 ≤ c 1 ≤ c 2 ≤ … and a closed discrete set T ⊆ R n , any C ∞ function g : R n → R can be approximated by an entire function f so that for k = 0 , 1 , 2 , …  , for all x ∈ R n such that | x | ≥ c k , and for each multi-index α such that | α | ≤ k , (a) | ( D α f ) ( x ) − ( D α g ) ( x ) | < ε ( x ) ; (b) ( D α f ) ( x ) = ( D α g ) ( x ) if x ∈ T . In this paper, we show that if C ⊆ R n + 1 is meager, A ⊆ R n is countable and disjoint from T , and for each multi-index α and p ∈ A we are given a countable dense set A p , α ⊆ R , then we can require also that (c) ( D α f ) ( p ) ∈ A p , α for p ∈ A and α any multi-index; (d) if x ∉ T , q = ( D α f ) ( x ) and there are values of p ∈ A arbitrarily close to x for which q ∈ A p , α , then there are values of p ∈ A arbitrarily close to x for which q = ( D α f ) ( p ) ; (e) for each α , { x ∈ R n : ( x , ( D α f ) ( x ) ) ∈ C } is meager in R n . Clause (d) is a surjectivity property whose full statement in the text also allows for finding solutions in A to equations of the form q = h ⁎ ( x , ( D α f ) ( x ) ) under similar assumptions, where h ( x , y ) = ( x , h ⁎ ( x , y ) ) is one of countably many given fiber-preserving homeomorphisms of open subsets of R n + 1 ≅ R n × R . We also prove a weaker corresponding result with “meager” replaced by “Lebesgue null.” In this context, the approximating function is C ∞ rather than entire, and we do not know whether it can be taken to be entire. The first result builds on earlier work of the author which deduced special cases of it from forcing theorems using absoluteness arguments. The proofs here do not use forcing. [ABSTRACT FROM AUTHOR]

This paper investigates the structure of points u ∈ A N that are such that the omega-limit set ω ( u , σ ) is precisely X , where X ⊆ A N is an internally transitive shift space. We then use those results to study the possible structures of the omega-limit set of the turning point for a unimodal map. Examples are provided of unimodal maps f where no iterate of the turning point c is recurrent and ω ( c , f ) is either a minimal Cantor set or properly contains a minimal Cantor set. [ABSTRACT FROM AUTHOR]

Let M be a hyperbolizable, non-trivial compression body without toroidal boundary components. In this paper, we characterize which discrete and faithful representations of π 1 ( M ) into PSL ( 2 , C ) are separable-stable. The set of separable-stable representations forms a domain of discontinuity for the action of Out ( π 1 ( M ) ) on the PSL ( 2 , C ) -character variety of π 1 ( M ) . [ABSTRACT FROM AUTHOR]

The main purpose of the paper is the proof of a cardinal inequality for a space with points G δ , obtained with the help of a long version of the Menger game. This result improves a similar one of Scheepers and Tall. [ABSTRACT FROM AUTHOR]

For a topological space X , let Σ ( X ) be the family of all subrings of C ( X ) containing C ⁎ ( X ) . In 1997, S.K. Acharyya, K.C. Chattopadhyay and D.P. Ghosh showed that for an A ( X ) ∈ Σ ( X ) , the intersection of all free maximal ideals of A ( X ) is an ideal of C ∞ ( X ) containing C K ( X ) . In this paper we define z ∞ -ideals and nicely balanced ideals of C ∞ ( X ) and show that if I is a z ∞ -ideal of C ∞ ( X ) then I is the intersection of all free maximal ideals of A ( X ) for some A ( X ) ∈ Σ ( X ) if and only if I is nicely balanced. Examples related to z ∞ -ideals and nicely balanced ideals are given. [ABSTRACT FROM AUTHOR]

Common rim-finite spaces seem to always contain arcs and, in fact, Ward showed [4] that any non-trivial rim-finite continuum must contain an arc. However, the compactness condition cannot be removed completely. In [3] a non-compact non-zero-dimensional, rim-64, arc-free separable metric space is given. Hence there are natural numbers n for which rim- n , non-zero-dimensional, arc-free separable metric spaces exist. In this paper, we determine precisely which values of n allow this. For n ≥ 3 there are separable metric, non-zero-dimensional arc-free spaces which are rim- n (and not rim-( n − 1 )). We also prove that any non-zero-dimensional and rim-2 space must contain an arc, and any rim-1 space must necessarily be zero-dimensional. [ABSTRACT FROM AUTHOR]

Given a metric continuum X , let C ( X ) be the hyperspace of subcontinua of X and Cone ( X ) the topological cone of X . We say that a continuum X is cone-embeddable in C ( X ) provided that there is an embedding h from Cone ( X ) into C ( X ) such that h ( x , 0 ) = { x } for each x in X . In this paper, we present some results concerning compactifications X of rays, union of two rays, and real lines which are cone-embeddable in C ( X ) . [ABSTRACT FROM AUTHOR]

In the definition of a set-open topology on C ( X ) , the set of all real-valued continuous functions on a Tychonoff space X , we use a certain family of subsets of X and open subsets of R . But instead of using this traditional way to define topologies on C ( X ) , in this paper, we adopt a different approach to define two interesting topologies on C ( X ) . We call them the open-point and the bi-point-open topologies and study the separation and countability properties of these topologies. [ABSTRACT FROM AUTHOR]

In this paper, we will give a sufficient condition for self-amalgamation of a strongly irreducible Heegaard splitting along any essential subsurface to be unstabilized. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish relationships between Lebesque and inductive dimensions ( m , n ) -dim and ( m , n ) -Ind. The main results are the following: 1. For every normal space X we have ( m , n ) - dim X = ⌊ dim X n ⌋ . 2. For every metrizable space X we have ( m , n ) - dim X = ( m , n ) - Ind X . [ABSTRACT FROM AUTHOR]

In this paper, we will investigate how radiality occurs in topological spaces by considering neighbourhood bases generated by nests. We will define a new subclass of radial spaces that contains LOTS, GO-spaces and spaces with well-ordered neighbourhood bases, called the independently-based spaces. We show that first-countable spaces are precisely the independently-based, strongly Fréchet spaces and we give an example of a Fréchet–Urysohn space that is neither independently-based nor strongly Fréchet. [ABSTRACT FROM AUTHOR]