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On the dichotomy of a locally compact semitopological monoid of order isomorphisms between principal filters of $\mathbb{N}^n$ with adjoined zeroAug 22 2019Let $n$ be any positive integer and $\mathscr{I\!P\!F}(\mathbb{N}^n)$ be the semigroup of all order isomorphisms between principal filters of the $n$-th power of the set of positive integers $\mathbb{N}$ with the product order. We prove that a Hausdorff ... More

The doubling metric and doubling measuresAug 20 2019We introduce the so--called doubling metric on the collection of non--empty bounded open subsets of a metric space. Given a subset $U$ of a metric space $X$, the predecessor $U_{*}$ of $U$ is defined by doubling the radii of all open balls contained inside ... More

Transversal, $T_{1}$-independent, and $T_{1}$-complementary paratopological group topologiesAug 17 2019We discuss the class of paratopological groups which admits a transversal, $T_{1}$-independent and $T_{1}$-complementary paratopological group topology. We show that the Sorgenfrey line does not admit a $T_{1}$-complementary Hausdorff paratopological ... More

Uniqueness of the hyperspaces $C(p,X)$ in the class of treesAug 16 2019Given a continuum $X$ and $p\in X$, we will consider the hyperspace $C(p,X)$ of all subcontinua of $X$ containing $p$. Given a family of continua $\mathcal{C}$, a continuum $X\in\mathcal{C}$ and $p\in X$, we say that $(X,p)$ has unique hyperspace $C(p,X)$ ... More

The hyperspaces HS(p,X)Aug 16 2019Let $X$ be a continuum and let $C(X)$ denote the hyperspace of subcontinua of $X$, endowed with the Hausdorff metric. For $p\in X$, define the hyperspace $C(p,X)=\{A\in C(X):p\in A\}$ as a subspace of $C(X)$. In this paper we introduced the quotient space ... More

The hyperspaces $HS(p,X)$Aug 16 2019Aug 20 2019Let $X$ be a continuum and let $C(X)$ denote the hyperspace of subcontinua of $X$, endowed with the Hausdorff metric. For $p\in X$, define the hyperspace $C(p,X)=\{A\in C(X):p\in A\}$ as a subspace of $C(X)$. In this paper we introduced the quotient space ... More

A finitary structure theorem for vertex-transitive graphs of polynomial growthAug 16 2019We prove a quantitative, finitary version of Trofimov's result that a connected, locally finite vertex-transitive graph G of polynomial growth admits a quotient with finite fibres on which the action of Aut(G) is virtually nilpotent with finite vertex ... More

Cardinal invariants of Haar null and Haar meager setsAug 15 2019Aug 19 2019A subset $X$ of a Polish group $G$ is \emph{Haar null} if there exists a Borel probability measure $\mu$ and a Borel set $B$ containing $X$ such that $\mu(gBh)=0$ for every $g,h \in G$. A set $X$ is \emph{Haar meager} if there exists a compact metric ... More

Cardinal invariants of Haar null and Haar meager setsAug 15 2019A ssubset $X$ of a Polish group $G$ is \emph{Haar null} if there exists a Borel probability measure $\mu$ and a Borel set $B$ containing $X$ such that $\mu(gBh)=0$ for every $g,h \in G$. A set $X$ is \emph{Haar meager} if there exists a compact metric ... More

A remark on locally direct product subsets in a topological Cartesian spaceAug 15 2019Let $X$ and $Y$ be topological spaces. Let $C$ be a path-connected closed set of $X\times Y$. Suppose that $C$ is locally direct product, that is, for any $(a,b)\in X\times Y$, there exist an open set $U$ of $X$, an open set $V$ of $Y$, a subset $I$ of ... More

A Tauberian theorem for ideal statistical convergenceAug 13 2019Given an ideal $\mathcal{I}$ on the positive integers, a real sequence $(x_n)$ is said to be $\mathcal{I}$-statistically convergent to $\ell$ provided that $$ \textstyle \left\{n \in \mathbf{N}: \frac{1}{n}|\{k \le n: x_k \notin U\}| \ge \varepsilon\right\} ... More

On the lattice of weak topologies on the bicyclic monoid with adjoined zeroAug 13 2019A Hausdorff topology $\tau$ on the bicyclic monoid $\mathcal{C}^0$ is called {\em weak} if it is contained in the coarsest inverse semigroup topology on $\mathcal{C}^0$. We show that the lattice $\mathcal{W}$ of all weak shift-continuous topologies on ... More

On Reeb graphs induced from smooth functions on closed or open surfacesAug 12 2019Aug 21 2019For a smooth function on a smooth manifold of a suitable class, the space of all the connected components of inverse images is the graph and called the Reeb graph. Reeb graphs are fundamental tools in the algebraic and differential topological theory ... More

On Reeb graphs induced from smooth functions on closed or open surfacesAug 12 2019For a smooth function on a smooth manifold of a suitable class, the space of all the connected components of inverse images is the graph and called the Reeb graph. Reeb graphs are fundamental tools in the algebraic and differential topological theory ... More

Compactly generated spaces and quasi-spaces in topologyAug 12 2019The notions of compactness and Hausdorff separation for generalized enriched categories allow us, as classically done for the category $\mathsf{Top}$ of topological spaces and continuous functions, to study $\textit{compactly generated spaces}$ and $\textit{quasi-spaces}$ ... More

Decompositions of set-valued mappingsAug 11 2019Let $X$ be a set, $B_{X}$ denotes the family of all subsets of $X$ and $F: X \longrightarrow B_{X}$ be a set-valued mapping such that $x \in F(x)$, $sup _{x\in X} F(x)< \kappa$, $sup _{x\in X} F^{-1}(x)< \kappa$ for all $x\in X$ and some infinite cardinal ... More

On convergent sequences in dual groupsAug 09 2019We provide some characterizations of precompact abelian groups $G$ whose dual group $G_p^\wedge$ endowed with the pointwise convergence topology on elements of $G$ contains a nontrivial convergent sequence. In the special case of precompact abelian \emph{torsion} ... More

Structure of Finite-Dimensional ProtoriAug 08 2019A Structure Theorem for Protori is derived for the category of finite-dimensional protori(compact connected abelian groups), which details the interplay between the properties of density, discreteness, torsion, and divisibility within a finite-dimensional ... More

Hyperspaces of infinite compacta with finitely many accumulation pointsAug 07 2019Vietoris hyperspaces $\mathcal A_n(X)$ ($\mathcal A_\omega(X)$) of infinite compact subsets of a metric space $X$ which have at most $n$ (finitely many, resp.) accumulation points are studied. If $X$ is a dense-in-itself, 0-dimensional Polish space, then ... More

Some comments on Laakso graphs and sets of differencesAug 07 2019We recall a variation of a construction due to Laakso \cite{LA}, also used by Lang and Plaut \cite{LA} of a doubling metric space $X$ that cannot be embedded into any Hilbert space. We give a more concrete version of this construction and motivated by ... More

Rough $I$-convergence in cone metric spacesAug 06 2019Here we have studied the notion of rough $I$-convergence as an extension of the idea of rough convergence in a cone metric space using ideals. We have further introduced the notion of rough $I^*$-convergence of sequences in a cone metric space to find ... More

Self-injectivity of $\EuScript{M}(X,\mathcal{A})$ versus $\EuScript{M}(X,\mathcal{A})$ modulo its socleAug 02 2019Let $\mathcal{A}$ be a field of subsets of a set $X$ and $\EuScript{M}(X,\mathcal{A})$ be the ring of all real valued $\mathcal{A}$-measurable functions on $X$. It is shown that $\EuScript{M}(X,\mathcal{A})$ is self-injective if and only if $\mathcal{A}$ ... More

Linear operators with infinite entropyAug 01 2019We examine the chaotic behavior of certain continuous linear operators on infinite-dimensional Banach spaces, and provide several equivalent characterizations of when these operators have infinite topological entropy. For example, it is shown that infinite ... More

Applications of Bornological Covering Properties in Metric SpacesJul 31 2019Using the idea of strong uniform convergence on bornology, Caserta, Di Maio and Ko\v{c}inac studied open covers and selection principles in the realm of metric spaces (associated with a bornology) and function spaces (w.r.t. the topology of strong uniform ... More

Closed subsets of compact-like topological spacesJul 28 2019Aug 08 2019We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We prove that each Hausdorff topological space can be embedded as a closed subspace into an H-closed topological space. However, the semigroup ... More

Closed subsets of compact-like topological spacesJul 28 2019We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We prove that each Hausdorff topological space can be embedded as a closed subspace into an H-closed topological space. However, the semigroup ... More

Inverse systems with simplicial bonding maps and cell structuresJul 26 2019For a topologically complete space $X$ and a family of closed covers $\mathcal A$ of $X$ satisfying a "local refinement condition" and a "completeness condition," we give a construction of an inverse system $\mathbf{ N}_{\mathcal A}$ of simplicial complexes ... More

A note on compact-like semitopological groupsJul 25 2019The note contains a few results related to separation axioms and automatic continuity of operations in compact-like semitopological groups. In particular, is presented a semiregular semitopological group $G$ which is not $T_3$. We show that each weakly ... More

A note on compact-like semitopological groupsJul 25 2019Aug 07 2019The note contains a few results related to separation axioms and automatic continuity of operations in compact-like semitopological groups. In particular, is presented a semiregular semitopological group $G$ which is not $T_3$. We show that each weakly ... More

The Transversality on locally pseudocompact groupsJul 25 2019Two non-discrete Hausdorff group topologies $\tau, \delta$ on a group $G$ are called {\it transversal} if the least upper bound $\tau\vee \delta$ of $\tau$ and $\delta$ is the discrete topology. In this paper, we main discuss the transversality of locally ... More

Semi-simple groups of compact 16-dimensional planesJul 24 2019The automorphism group $\Sigma$ of a compact topological projective plane with a $16$-dimensional point space is a locally compact group. If the dimension of $\Sigma$ is at least $29$, then $\Sigma$ is known to be a Lie group. For the connected component ... More

Sheaf theoretic characterization of topological etale groupoidsJul 24 2019A sheaf can be defined in two ways. As the etale space, or as the contravariant functor. In this paper, by analogy, we characterize the topological etale groupoids by a sheaf theoritic way. For that purpose, we introduce a pseudogroup sheaf as generalization ... More

Splitting chains, tunnels and twisted sumsJul 23 2019We study splitting chains in $\mathscr{P}(\omega)$, that is, families of subsets of $\omega$ which are linearly ordered by $\subseteq^*$ and which are splitting. We prove that their existence is independent of axioms of $\mathsf{ZFC}$. We show that they ... More

Inductive dimensions of coarse proximity spacesJul 23 2019In this paper, we define the asymptotic inductive dimension, $asInd$, of coarse proximity spaces. In the case of metric spaces equipped with their metric coarse proximity structure, this definition is equivalent to the definition of $asInd$ given by Dranishnikov ... More

$Ps$-normal and $Ps$-Tychonoff spacesJul 20 2019A space $X$ is called $Ps$-normal($Ps$-Tychonoff) space if there exists a normal(Tychonoff) space $Y$ and a bijection $f: X\mapsto Y$ such that $f\lvert_K:K\mapsto f(K)$ is homeomorphism for any pseudocompact subset $K$ of $X$. We establish a few relations ... More

Virtual Parity Alexander PolynomialJul 19 2019In this paper, we define the parity virtual Alexander polynomial following the work of BDGGHN [1] and Kaestner and Kauffman [10]. The properties of this invariant are explored and some examples are computed. In particular, the invariant demonstrates that ... More

Universal sets for idealsJul 19 2019In this paper we consider a notion of universal sets for ideals. We show that there exist universal sets of minimal Borel complexity for classic ideals like null subsets of $2^\omega$ and meager subsets of any Polish space, and demonstrate that the existence ... More

On some relations between properties of invariant $σ$-ideals in Polish spacesJul 18 2019In this paper we shall consider a couple of properties of $\sigma$-ideals and study relations between them. Namely we will prove that $\mathfrak{c}$-cc $\sigma$-ideals are tall and that the Weaker Smital Property implies that every Borel $\mathcal{I}$-positive ... More

A note on sets avoiding rational distancesJul 18 2019In this paper we shall give a short proof of the result originally obtained by Ashutosh Kumar that for each $A\subset \mathbb{R}$ there exists $B\subset A$ full in $A$ such that no distance between two distinct points from $B$ is rational. We will construct ... More

The closure-complement-frontier problem in saturated polytopological spacesJul 18 2019Let $X$ be a space equipped with $n$ topologies $\tau_1,...,\tau_n$ which are pairwise comparable and saturated, and for each $1\leq i\leq n$ let $k_i$ and $f_i$ be the associated topological closure and frontier operators, respectively. Inspired by the ... More

The closure-complement-frontier problem in saturated polytopological spacesJul 18 2019Aug 03 2019Let $X$ be a space equipped with $n$ topologies $\tau_1,...,\tau_n$ which are pairwise comparable and saturated, and for each $1\leq i\leq n$ let $k_i$ and $f_i$ be the associated topological closure and frontier operators, respectively. Inspired by the ... More

Optimization Of Quasi-convex Function Over Product Measure SetsJul 18 2019We consider a generalization of the Bauer maximum principle. We work with tensorial products of convex measures sets, that are non necessarily compact but generated by their extreme points. We show that the maximum of a quasi-convex lower semicontinuous ... More

A conjecture on the lengths of filling pairsJul 16 2019A pair $(\alpha, \beta)$ of simple closed geodesics on a closed and oriented hyperbolic surface $M_g$ of genus $g$ is called a filling pair if the complementary components of $\alpha\cup\beta$ in $M_g$ are simply connected. The length of a filling pair ... More

A conjecture on the lengths of filling pairsJul 16 2019Aug 21 2019A pair $(\alpha, \beta)$ of simple closed geodesics on a closed and oriented hyperbolic surface $M_g$ of genus $g$ is called a filling pair if the complementary components of $\alpha\cup\beta$ in $M_g$ are simply connected. The length of a filling pair ... More

On some kinds of weakly sober spacesJul 15 2019In \cite{E_2018}, Ern\'e relaxed the concept of sobriety in order to extend the theory of sober spaces and locally hypercompact spaces to situations where directed joins were missing, and introduced three kinds of non-sober spaces: cut spaces, weakly ... More

Topological realizations of groups in Alexandroff spacesJul 12 2019Given a group $G$, we provide a constructive method to get infinitely many (non-homotopy-equivalent) Alexandroff spaces, such that the group of autohomeomorphisms, the group of homotopy classes of self-homotopy equivalences and the pointed version are ... More

Constructing Wadge classesJul 12 2019We show that, assuming the Axiom of Determinacy, every non-selfdual Wadge class can be constructed by starting with those of level $\omega_1$ (that is, the ones that are closed under Borel preimages) and iteratively applying the operations of expansion ... More

On Ramsey properties, function spaces, and topological gamesJul 11 2019An open question of Gruenhage asks if all strategically selectively separable spaces are Markov selectively separable, a game-theoretic statement known to hold for countable spaces. As a corollary of a result by Berner and Juh$\acute{a}$sz, we note that ... More

Artin glueings of frames as semidirect productsJul 11 2019Artin glueings provide a way to reconstruct a frame from a closed sublocale and its open complement. We show that Artin glueings can be described as split extensions satisfying a Schreier-type condition in the category frames with finite-meet preserving ... More

Core-compactness of Smyth powerspacesJul 10 2019We prove that the Smyth powerspace Q(X) of a topological space X is core-compact if and only if X is locally compact. As a straightforward consequence we obtain that the Smyth powerspace construction does not preserve core-compactness generally.

A common extension of Arhangel'skii's Theorem and the Hajnal-Juhasz inequalityJul 09 2019We present a bound for the weak Lindel\"of number of the $G_\delta$-modification of a Hausdorff space which implies various known cardinal inequalities, including the following two fundamental results in the theory of cardinal invariants in topology: ... More

Hairy Cantor setsJul 07 2019We introduce a topological object, called hairy Cantor set, which in many ways enjoys the universal features of objects like Jordan curve, Cantor set, Cantor bouquet, hairy Jordan curve, etc. We give an axiomatic characterisation of hairy Cantor sets, ... More

Generic homeomorphisms with shadowing of one-dimensional continuaJul 04 2019In this article we show that there are homeomorphisms of plane continua whose conjugacy class is residual and have the shadowing property.

Cardinal-indexed classifying spaces for families of subgroups of any topological groupJul 01 2019Any $G$-space isovariantly or approximately covered by tubes is the pullback of a classifying space indexed by the orbit types of the tubes and the cardinality of the cover.

A note on weak-star and norm Borel sets in the dual of the space of continuous functionsJul 01 2019Let $Bo(T,\tau)$ be the Borel $\sigma$-algebra generated by the topology $\tau$ on $T$. In this paper we show that if $K$ is a Hausdorff compact space, then every subset of $K$ is a Borel set if, and only if, $$Bo(C^*(K),w^*)=Bo(C^*(K),\|\cdot\|);$$ where ... More

The two digital homology theoriesJun 30 2019In this paper we prove results relating to four homology theories developed in the topology of digital images: a simplicial homology theory by Arslan et al which is the homology of the clique complex, a singular simplicial homology theory by Lee, a cubical ... More

A new approach to coincidence and common fixed points under a homotopy of families of mappings in $b$-metric spacesJun 30 2019In this paper we derive coincidence and common fixed point results under order homotopies of families of mappings in preordered $b$-metric spaces.

Statistical convergence of nets through directed setsJun 28 2019The concept of statistical convergence based on asymptotic density is introduced in this article through nets. Some possible extensions of classical results for statistical convergence of sequences are obtained in this article, with extensions to nets. ... More

On a metric on the space of monetary risk measuresJun 26 2019We introduce a metric on the space of monetary risk measure, which generates the point-wise convergence topology and extends the metric on the initial compactum.

Locally ordered topological spacesJun 26 2019While topology given by a linear order has been extensively studied, this cannot be said about the case when the order is given only locally. Our aim in this paper is to fill this gap. We consider relation between local orderability and separation axioms ... More

Existence of well-filterifications of $T_0$ topological spacesJun 26 2019We prove that for every $T_0$ space $X$, there is a well-filtered space $W(X)$ and a continuous mapping $\eta_X: X\lra W(X)$ such that for any well-filtered space $Y$ and any continuous mapping $f: X\lra Y$ there is a unique continuous mapping $\hat{f}: ... More

Existence of well-filterifications of $T_0$ topological spacesJun 26 2019Jul 15 2019We prove that for every $T_0$ space $X$, there is a well-filtered space $W(X)$ and a continuous mapping $\eta_X: X\lra W(X)$ such that for any well-filtered space $Y$ and any continuous mapping $f: X\lra Y$ there is a unique continuous mapping $\hat{f}: ... More

Topological reducibilities for discontinuous functions and their structuresJun 25 2019In this article, we give a full description of a topological many-one degree structure of real-valued functions, recently introduced by Day-Downey-Westrick. We also point out that their characterization of the Bourgain rank of a Baire-one function of ... More

Combinatorial properties on nodec countable spaces with analytic topologyJun 25 2019We study some variations of the product topology on families of clopen subsets of $2^{\mathbb{N}}\times\mathbb{N}$ in order to construct countable nodec regular spaces (i.e. in which every nowhere dense set is closed) with analytic topology which in addition ... More

The Golomb topology on a Dedekind domain and the group of units of its quotientsJun 24 2019We study the Golomb spaces of Dedekind domains with torsion class group. In particular, we show that a homeomorphism between two such spaces sends prime ideals into prime ideals and preserves the $P$-adic topology on $R\setminus P$. Under certain hypothesis, ... More

Random subgroups, automorphisms, splittingsJun 23 2019We show that, if $H$ is a random subgroup of a finitely generated free group $F_k$, only inner automorphisms of $F_k$ may leave $H$ invariant. A similar result holds for random subgroups of toral relatively hyperbolic groups, more generally of groups ... More

A T_0 Compactification Of A Tychonoff Space Using The Rings Of Baire One FunctionsJun 20 2019In this article, we continue our study of Baire one functions on a topological space $X$, denoted by $B_1(X)$ and extend the well known M. H. Stones's theorem from $C(X)$ to $B_1(X)$. Introducing the structure space of $B_1(X)$, it is observed that $X$ ... More

Extensions of semigroups by symmetric inverse semigroups of a bounded finite rankJun 19 2019We study the semigroup extension $\mathscr{I}_\lambda^n(S)$ of a semigroup $S$ by symmetric inverse semigroups of a bounded finite rank. We describe idempotents and regular elements of the semigroups $\mathscr{I}_\lambda^n(S)$ and $\overline{\mathscr{I}_\lambda^n}(S)$ ... More

Suprema in spectral spaces and the constructible closureJun 17 2019Given an arbitrary spectral space $X$, we endow it with its specialization order $\leq$ and we study the interplay between suprema of subsets of $(X,\leq)$ and the constructible topology. More precisely, we provide sufficient conditions in order for the ... More

On infinite iterations of the functor of idempotent probability measuresJun 17 2019In this paper we establish that the functor of idempotent probability measures acting in the category of compacta and their continous mappings is perfect metrisable

Sequences of contractions on cone metric spaces over Banach algebras and applications to nonlinear systems of equations and systems of differential equationsJun 14 2019It is well known that fixed point problems of contractive-type mappings defined on cone metric spaces over Banach algebras are not equivalent to those in usual metric spaces (see [3] and [10]). In this framework, the novelty of the present paper represents ... More

Group geometrical axioms for magic states of quantum computingJun 14 2019Let $H$ be a non trivial subgroup of index $d$ of a free group $G$ and $N$ the normal closure of $H$ in $G$. The coset organization in a subgroup $H$ of $G$ provides a group $P$ of permutation gates whose common eigenstates are either stabilizer states ... More

Extensions of dualities and a new approach to the de Vries dualityJun 13 2019We prove a general categorical theorem for the extension of dualities. Applying it, we present new proofs of the de Vries Duality Theorem for the category CHaus of compact Hausdorff spaces and continuous maps, and of the recent Bezhanishvili-Morandi-Olberding ... More

Embeddings into countably compact Hausdorff spacesJun 11 2019In this paper we consider the problem of characterization of topological spaces that embed into countably compact Hausdorff spaces. We study the separation axioms of subspaces of countably compact Hausdorff spaces and construct an example of a regular ... More

The Frame of Nuclei of an Alexandroff SpaceJun 09 2019Let $\mathcal{O}S$ be the frame of open sets of a topological space $S$, and let $N(\mathcal{O}S)$ be the frame of nuclei of $\mathcal{O}S$. For an Alexandroff space $S$, we prove that $N(\mathcal{O}S)$ is spatial iff the infinite binary tree $\mathscr ... More

When is the frame of nuclei spatial: A new approachJun 09 2019For a frame $L$, let $X_L$ be the Esakia space of $L$. We identify a special subset $Y_L$ of $X_L$ consisting of nuclear points of $X_L$, and prove the following results: $L$ is spatial iff $Y_L$ is dense in $X_L$. If $L$ is spatial, then $N(L)$ is spatial ... More

Supercompact minus compact is superJun 09 2019According to a folklore characterization of supercompact spaces, a compact Hausdorff space is supercompact if and only if it has a binary closed $k$-network. This characterization suggests to call a topological space $super$ if it has a binary closed ... More

Asymptotic property C of the countable direct sum of uniformly discrete $0$-hyperbolic spacesJun 08 2019We define the direct sum of a countable family of pointed metric spaces in a way resembling the direct sum of groups. Then we prove that if a family consists of $0$-hyperbolic (in the sense of Gromov) and $D$-discrete spaces, then its direct sum has asymptotic ... More

Compactness properties defined by open-point gamesJun 07 2019Let 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 n'th move, Player A chooses ... More

Triple extension of Tietze theorem and Baer criterionJun 06 2019In this paper, through the combination of Tietze extension theorem and Baer criteria, we build a new mathematical structure which is similar to a triangular pyramid, and then we prove that the topological space which we call it Tb appeared as a result ... More

The route to chaos in routing games: Population increase drives period-doubling instability, chaos & inefficiency with Price of Anarchy equal to oneJun 06 2019We study a learning dynamic model of routing (congestion) games to explore how an increase in the total demand influences system performance. We focus on non-atomic routing games with two parallel edges of linear cost, where all agents evolve using Multiplicative ... More

Remarks on GP-metric and partial metric spaces and fixed points resultsJun 02 2019In this paper we give some relationship between G-metric spaces, partial metric spaces and GP-metric spaces.

Embedding topological spaces into Hausdorff $ω$-bounded spacesJun 01 2019We discuss the problem of embeddibility of a topological space into a Hausdorff $\omega$-bounded space, and present two canonical constructions of such an embedding.

On a typical compact set as the attractor of generalized iterated function systems of infinite orderMay 31 2019In 2013 Balka and M\'ath\'e showed that in uncountable polish spaces the typical compact set is not a fractal of any IFS. In 2008 Miculescu and Mihail introduced a concept of a generalized iterated function system (GIFS in short), a particular extension ... More

A short proof of the metrizability of $\mathcal{F}$-metric spacesMay 29 2019The main purpose of this manuscript is to provide a short proof of the metrizability of $\mathcal{F}$-metric spaces introduced by Jleli and Samet in \cite[\, Jleli, M. and Samet, B., On a new generalization of metric spaces, J. Fixed Point Theory Appl. ... More

Point mutations in a growing cell populationMay 29 2019A growing cell population such as cancer or bacteria accumulates point mutations (single nucleotide changes in DNA). To model this process, we consider that cells divide and die as a branching process, and that each cell contains a sequence of nucleotides ... More

Soft N-Topological SpacesMay 29 2019Very recently, the idea of studying structures equipped with two or more soft topologies has been considered by several researchers. Soft bitopological spaces were introduced and studied, in 2014, by Ittanagi as a soft counterpart of the notion of bitopological ... More

A Soft Embedding Lemma for Soft Topological SpacesMay 29 2019In 1999, Molodtsov initiated the theory of soft sets as a new mathematical tool for dealing with uncertainties in many fields of applied sciences. In 2011, Shabir and Naz introduced and studied the notion of soft topological spaces, also defining and ... More

On $\mathcal{H}_Y$-IdealsMay 28 2019In this article, we continue the studying of $\mathcal{H}_Y$-ideals. We introducing two notions fixed and free $\mathcal{H}_Y$-ideals as an extension of fixed and free z-ideals in C(X) and relative $\mathcal{H}_Y$-ideals as an extension of relative z-ideals. ... More

On $\mathcal{H}_Y$-IdealsMay 28 2019Jun 08 2019In this article, we continue the studying of $\mathcal{H}_Y$-ideals. We introducing two notions fixed and free $\mathcal{H}_Y$-ideals as an extension of fixed and free z-ideals in C(X) and relative $\mathcal{H}_Y$-ideals as an extension of relative z-ideals. ... More

Selectors for dense subsets of function spacesMay 24 2019Jul 01 2019Let $USC^*_p(X)$ be the topological space of real upper semicontinuous bounded functions defined on $X$ with the subspace topology of the product topology on ${}^X\mathbb{R}$. $\tilde\Phi^{\uparrow},\tilde\Psi^{\uparrow}$ are the sets of all upper sequentially ... More

Selectors for dense subsets of function spacesMay 24 2019Let $USC^*_p(X)$ be the topological space of real upper semicontinuous bounded functions defined on $X$ with the subspace topology of the product topology on ${}^X\mathbb{R}$. $\tilde\Phi^{\uparrow},\tilde\Psi^{\uparrow}$ are the sets of all upper sequentially ... More

Countable approximation of topological $G$-manifolds, III: arbitrary Lie groups $G$May 24 2019The Hilbert--Smith conjecture states, for any connected topological manifold $M$, any locally compact subgroup of $\mathrm{Homeo}(M)$ is a Lie group. We generalize basic results of Dugundji--Abels (1.8), Segal--Kosniowski--tomDieck (2.6), G Bredon (3.7), ... More

The continuous $d$-open homomorphism images and subgroups of $\mathbb{R}$-factorizabile paratopological groupsMay 23 2019In this paper, we prove that: (1) Let $f:G\rightarrow H$ be a continuous $d$-open surjective homomorphism; if $G$ is an $\mathbb{R}$-factorizabile paratopological group, then so is $H$. Peng and Zhang's result \cite[Theorem 1.7]{PZ} is improved. (2) Let ... More

The continuous $d$-open homomorphism images and subgroups of $\mathbb{R}$-factorizabile paratopological groupsMay 23 2019Jun 05 2019In this paper, we prove that: (1) Let $f:G\rightarrow H$ be a continuous $d$-open surjective homomorphism; if $G$ is an $\mathbb{R}$-factorizabile paratopological group, then so is $H$. Peng and Zhang's result \cite[Theorem 1.7]{PZ} is improved. (2) Let ... More

Finiteness of topological entropy for locally compact abelian groupsMay 23 2019We study the locally compact abelian groups in the class $\mathfrak E_{<\infty}$, that is, having only continuous endomorphisms of finite topological entropy, and in its subclass $\mathfrak E_0$, that is, having all continuous endomorphisms with vanishing ... More

On additive property of finitely additive measuresMay 22 2019By the additive property, we mean a condition under which $L^p$ spaces over finitely additive measures are complete. Basile and Rao gives a necessary and sufficient condition that a finite sum of finitely additive measures has the additive property. We ... More

Mycielski among treesMay 22 2019Two-dimensional version of the classical Mycielski theorem says that for every comeager or conull set $X\subseteq [0,1]^2$ there exists a perfect set $P\subseteq [0,1]$ such that $P\times P\subseteq X\cup \Delta$. We consider generalizations of this theorem ... More

A basic framework for fixed point theorems: ball spaces and spherical completenessMay 22 2019We systematically develop a general framework in\linebreak which various notions of functions being contractive, as well as of spaces being complete, can be simultaneously encoded. Derived from the notions of ultrametric balls and spherical completeness, ... More

Fine structure of the homomorphisms of the lattice of uniformly continuous functions on the lineMay 22 2019We provide a representation of the homomorphisms $U\longrightarrow \mathbb R$, where $U$ is the lattice of all uniformly continuous on the line. The resulting picture is sharp enough to describe the fine topological structure of the space of such homomorphisms. ... More