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 to be path-connected. We define the notion of a path-component base and show that if f admits a path-component base then is not path-connected, and we give a characterisation of path-connected

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

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

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

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 [4] we extended this duality to completely regular spaces. In this article we use this extension to characterize normal, Lindëlof, 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

In this paper, we study some new selection principles using the star operator which was introduced by Bal and Bhowmik (2017) [7]. If the Alexandroff duplicate A(X) of a space X has the property U1⋆(O,O) (Ufin⋆(O,O)), then X has the property U1⋆(O,O) (Ufin⋆(O,O)). If a space has the property Ufin⋆(O,O), then its inverse image under perfect open map has the property Ufin⋆(O,O). The product of a space

Well-filtered spaces play an increasing crucial role in domain theory. A recent development concerning well-filtered spaces is the establishment of the existence of well-filterification of T0-spaces. In this paper, we give yet another way for constructing the well-filterification of T0-spaces. Additionally, our construction gives a negative answer to an open problem posed by Xiaoquan Xu in [9].

In this paper we use the equivariant theory of retracts to give some results about the topological social choice theory. One of them is an equivariant version of the Chichilnisky-Heal Impossibility Theorem when a compact group acts on the preferences space. We also improve one result about the preservation of equivariant ANR's by symmetric powers, which is related to a geometrical characterization

We inductively define layers of colorings of knot and knotted surface diagrams using ternary quasigroups. Homological invariants from such systems of colorings use shorter differentials and of higher degree than the standard homology differentials, and give access to typically more complex homology groups.

Continuous functions over compact Hausdorff spaces have been completely characterised. We consider the more general problem: given a set-valued function T from an arbitrary set X to itself, does there exist a compact Hausdorff topology on X with respect to which T is upper semicontinuous? We give conditions that are necessary for T to be upper semicontinuous and point-closed if X is a compact Hausdorff

We show, in contrast to the cases of complex and quaternionic partial flag manifolds, the rational cohomology rings of the real and oriented partial flag manifolds are generated not only by the characteristic classes of their canonical vector bundles, but also by exterior algebras arising from the rational cohomology rings of real Stiefel manifolds. We also discuss the rational and the mod-2 equivariant

The Poincaré compactification is an extension of a polynomial vector field to a compact manifold. We generalize this construction to weight-homogeneous vector fields with weight exponent (1,ℓ) and different weight-degrees. Then we apply the new method to obtain some new information about the infinite singular points and in particular for Hamiltonian systems under generic conditions.

In previous work we showed that if a paracompact Hausdorff space contains a nontrivial component, then none of the Čech systems of the nerves of its open covers can be an approximate (inverse) system. This extended a theorem of the first author who, in answering a question of S. Mardešić, proved this result in the restricted case that the given Hausdorff space was arc-like (and hence is nontrivial

For a pair (G,X), where G is a topological group and X is a subspace of G, we consider continuous mappings f of G onto a topological space Y such that f(G)=Y=f(X). If, in addition, X is compact (σ-compact), then Y is called a gc-image of G (respectively, a gσc-image of G). For example, all dyadic compacta are gc-images of compact topological groups. Below Esenin-Vol'pin's Theorem on metrizability of

Applying a general categorical construction for the extension of dualities, we present a new proof of the Fedorchuk duality between the category of compact Hausdorff spaces with their quasi-open mappings and the category of complete normal contact algebras with suprema-preserving Boolean homomorphisms which reflect the contact relation.

Let S be a topological property of sequences (such as, for example, “to contain a convergent subsequence” or “to have an accumulation point”). We introduce the following open-point game OP(X,S) on a topological space X. In the nth move, Player O chooses a non-empty open set Un⊆X, and Player P responds by selecting a point xn∈Un. Player P wins the game if the sequence {xn:n∈N} satisfies property S in

We answer questions about P-filters in the Cohen, random, and Laver forcing extensions of models of CH. In the case of the ℵ2-random real poset, we prove that if □ℵ1 also holds in the ground model, then there are P-points of ω⁎ in the extension. The majority of the paper investigates the question of whether ω⁎ can be covered by nowhere dense P-sets. We prove that this is not the case if ℵ2-Cohen reals

Let C be an epireflective subcategory of Top and let rC be the epireflective functor associated with C. If A denotes a (semi)topological algebraic subcategory of Top, we study when rC(A) is an epireflective subcategory of A. We prove that this is always the case for semi-topological structures and we find some sufficient conditions for topological algebraic structures. We also study when the epireflective

We survey discrete and continuous model-theoretic notions which have important connections to general topology. We present a self-contained exposition of several interactions between continuous logic and Cp-theory which have applications to a classification problem involving Banach spaces not including c0 or lp, following recent results obtained by P. Casazza and J. Iovino for compact continuous logics

The object of the paper are the regular topological spaces X in which there exists a metric d related to the topology in the following way: for every nonempty open subset U of X and for every ε>0 there exists a nonempty open subset V of U with d-diameter less than ε. It is shown that such a space X is pseudo-almost Čech complete if, and only if, it contains a dense completely metrizable subset (X is

f:X→Y is an Fσ-mapping if f maps Fσ-sets in X to Fσ-sets in Y and f−1 maps Fσ-sets in Y back to Fσ-sets in X. R.W. Hansell, J.E. Jayne, and C.A. Rogers under some assumptions proved that every Fσ-mapping of an absolute Souslin-F set onto an absolute Souslin-F set is piecewise closed. We give a similar result for Souslin-F subsets of completely Baire spaces.

Let Y be a subspace of the product space X=∏i∈IXi and f:Y→Z be a continuous mapping. We present several conditions on Y and Z which guarantee the existence of the smallest (by inclusion) set J⊆I such that f depends on J. This happens, for example, if Y=X and Z is a T0-space. We also show that such a set J⊆I exists if either Y is open and dense in X and Z is Hausdorff or Y contains the σ-product σX(a)⊆X

We call a continuous map f:X→Y nowhere constant if it is not constant on any non-empty open subset of its domain X. Clearly, this is equivalent with the assumption that every fiber f−1(y) of f is nowhere dense in X. We call the continuous map f:X→Y pseudo-open if for each nowhere dense Z⊂Y its inverse image f−1(Z) is nowhere dense in X. Clearly, if Y is crowded, i.e. has no isolated points, then f

Given a family of metric continua {Xα:α∈J}, we consider the following property for the product X=∏α∈JXα: if M is a subcontinuum of X projecting onto each factor space, then M has arbitrarily small connected open neighborhoods. This property has been called fupcon (full projections imply connected open neighborhoods) property and it has been studied by several authors. Particularly, in 2018, A. Illanes

We prove a selection theorem for convex-valued lower semicontinuous mappings F to Fréchet spaces under the assumption that every value of F contains all interior (in the convex sense) points of its own closure. This is an extension of E. Michael's theorem 3.1‴ in [7] for mappings to Banach spaces. The desired continuous single-valued selection is constructed as a pointwise barycenter mapping with respect

We show that C(X) admits an equivalent pointwise lower semicontinuous locally uniformly rotund norm provided X is a Fedorchuk compact of spectral height 3. In other words, X admits a fully closed map f onto a metric compact Y such that f−1(y) is metrizable for all y∈Y. A continuous map of compacts f:X→Y is said to be fully closed if the intersection f(A)∩f(B) is finite for any disjoint closed subsets

For a subset A of the real line R, the Hattori space H(A) is a topological space whose underlying point set is the real R and whose topology is defined as follows: points from A possess the usual Euclidean neighbourhoods while remaining points are given the neighbourhoods of the Sorgenfrey line. We consider the linear homeomorphisms between the spaces Cp(H(A)), Cp(H(B)) and Cp(S), where H(A) and H(B)

A.V. Arhangel'skii introduced the dimension-like function Dind (see [2]) and some of its properties were proved in [1] and [5]. In this paper, we define the base dimension-like function of the type Dind, denoted by b-Dind, and study the property of universality for this function, proving that in a class of bases which is determined by b-Dind there exist universal elements.

For a Tychonoff space X and a family λ of subsets of X, we denote by Cλ(X) the T1-space of all real-valued continuous functions on X with the λ-open topology. A topological space is productively Lindelöf if its product with every Lindelöf space is Lindelöf. A space is indestructibly productively Lindelöf if it is productively Lindelöf in any extension by countably closed forcing. A Menger space is

Under Martin Axiom, we prove that for each ordinal γ<ω1 there exists a thin ultrafilter that belongs to the class Pγ of the P-hierarchy of ultrafilters. Since the class P2 of ultrafilters coincides with the class of P-points, this result generalizes a theorem of Flašková, which states that, under the Martin Axiom for countable posets, there exists a thin ultrafilter which is not a P-point. It is also

Assume that a functionally Hausdorff space X is a continuous image of a Čech complete space P with Lindelöf number l(P)

The aim of this paper is to investigate the class of quasi κ-metrizable spaces. This class is invariant with respect to arbitrary products and contains Shchepin's [8] κ-metrizable spaces as a proper subclass.

For a Tychonoff space X, the Dieudonné τ-completion of X, denoted by μτX, is investigated. The space μτX is defined as the completion of X with respect to the uniformity uXτ, where uXτ is generated by all continuous mappings of X to metric spaces of weight ≤τ. It is proved that a dense subspace Y of X is Pτ-(equivalently, Pτ⁎-)embedded in X iff uXτ|Y=uYτ iff Y⊂X⊂μτX. In this case, the free topological

We solve a problem posed by Tkachuk and Wilson, Question 5.10 in [16], on whether every first countable cellular-compact space is weakly Lindelöf. We actually obtain a stronger result and, as a by-product of it, we present a somewhat different proof of Tkachuk and Wilson theorem on the cardinality of a first countable cellular-compact space, valid for the wider class of Urysohn spaces. Moreover, our

In 1994, John Cobb asked: given N>m>k>0, does there exist a Cantor set in RN such that each of its projections into m-planes is exactly k-dimensional? Such sets were described for (N,m,k)=(2,1,1) by L. Antoine (1924) and for (N,m,m) by K. Borsuk (1947). Examples were constructed for the cases (3,2,1) by J. Cobb (1994), for (N,m,m−1) and in a different way for (N,N−1,N−2) by O. Frolkina (2010, 2019)

P.S. Alexandroff's idea is one of the three important sources of the theory of generalized metric spaces, which leads to classifying spaces by mappings. As one of the main methods of mutual classifications of spaces and mappings, the systems of covers of topological spaces have been accepted widely. In this paper, by sequences of certain cs-covers for spaces, we seek to the characterizations of spaces

In this paper, we introduce paratopogical left-loops, as natural generalizations of paratopogical left Bol loops defined by Z. Cai, S. Lin and W. He in [7], and prove that some known results for paratopological groups are still valid for paratopogical left-loops. The main results are: (i) every weakly first-countable paratopological left-loop is first-countable; (ii) a sequential, csf-countable, regular

We study analytic sets which does not contain closed subspaces of the first category in itself in connection with some problems of the descriptive theory of sets and functions.

We study the star version of selectively ccc spaces. Relations of this class of spaces with some other existing selective properties are discussed. A few examples which make considered spaces different from other spaces are provided.

For some productive and closed hereditary classes C in Hausdorff topological spaces a cardinal κ will be obtained such that every continuous map on a limit of an inverse system in C into a given space can be factorized via a limit of a subsystem of cardinality less than κ. Large cardinals will be needed to get some general results.

The free topological vector space V(X) over a Tychonoff space X is a pair consisting of a topological vector space V(X) and a continuous mapping i=iX:X→V(X) such that every continuous mapping f from X to a topological vector space E gives rise to a unique continuous linear operator f‾:V(X)→E with f=f‾∘i. In this paper, the k-property, Fréchet-Urysohn property, κ-Fréchet-Urysohn property and countable

For a Tychonoff space X, we denote by Cp(X) the space of all real valued continuous functions with the topology of pointwise convergence. In this short note, we remark that if Cp(X) and Cp(Y) are linearly homeomorphic and X is a Menger (resp., Hurewicz) space with property (⁎), then Y is also a Menger (resp., Hurewicz) space.

If a Tychonoff space X has a point-countable network of cardinality ≤ℵ1 and Y is a normal dense subspace of Cp(X), then Y is collectionwise normal. If a Tychonoff space X has a point-countable network of cardinality ≤ℵ1, then Cp(Cp(X)) also has a point-countable network of cardinality ≤ℵ1. We prove these statements and obtain some corollaries on equivalence of normality and collectionwise normality

We prove that for a strictly increasing sequence (mn) of natural numbers and d∈N,d≥2 the subgroup G=∑n∈NZ(dmn) endowed with the topology induced by the product ∏n∈NZ(dmn) has no Mackey topology. In other words the supremum of all locally quasi–convex group topologies on G having as character group ∑n∈NZ(dmn)∧ has a strictly larger character group. As a consequence we obtain that also Z(N) with the

Let τ be an uncountable cardinal. We prove that if A is a cover of the Tychonoff cube Iτ such that |A|≤τ, then some element A∈A is not homeomorphic to a topological group.

We prove that a strong version of Chang's Conjecture together with [Formula: see text] implies there are no [Formula: see text]-Aronszajn trees.