One-point Set Research Articles

СHARACTERIZATION OF ONE-POINT SET OF DISCONTINUOUS OF SEPARATELY CONTINUOUS FUNCTIONS OF SEVERAL VARIABLE

Investigations of the discontinuity points set of separately continuous functions of two or many variables (i.e. functions that are continuous with respect to each variable) were started in Rene Baire's dissertation \cite{Baire} and these investigations have been continued and developed by many mathematicians.Investigations of the discontinuity points set of separately continuous functions of two or many variables (i.e. functions that are continuous with respect to each variable) were started in Rene Baire's dissertation \cite{Baire} and these investigations have been continued and developed by many mathematicians. Investigations of separately continuous functions and their analogs with one-point set of points of discontinuity are of particular interest. It was proved in \cite{p-filter} that the existence of separately continuous functions with given one-point set of points of discontinuity of $G_\delta$ type is closely related to the properties of $P$-filter, and the answer to this question is independent of $ZFC$. It was proved in the \cite{p-filter-many} that the existence of a strongly separately continuous function $f:X_1\times ...\times X_n\to\mathbb{R}$ on the product of arbitrary completely regular spaces $X_k$ with an one-point set $\{(x_1,...,x_n)\}$ of points of discontinuity where $x_k$ is non-isolated $G_\delta$-point in $X_k$, is equivalent to NCPF (Near Coherence of $P$-filters). Strongly separately continuous function of $n$ variables is a function that for any fixed one variable is continuous with respect to other variables. It is clear that for the function of two variables strong separate continuity is equivalent to the separate continuity. In general each strongly separately continuous functions is separately continuous. But the existence of strongly separately continuous function is stronger than the existence of separately continuous function. In this paper we consider question what is necessity and sufficiency for existence a separately continuous function on the product of arbitrary completely regular spaces $X_k$ with an one-point set $\{(x_1,...,x_n)\}$ of points of discontinuity where $x_k$ is non-isolated $G_\delta$-point in $X_k$. First we prove that for We prove that the existence of such function is equivalent to the fact that for any $n$ $P$-filters there exist two that are near coherent.

On meromorphic functions sharing three one-point or two-point sets CM

On meromorphic functions sharing three one-point or two-point sets CM

Finite graphs have unique second and third symmetric product suspension

Let X be a metric continuum and n be a positive integer. Consider the hyperspace Fn(X) of all nonempty closed subsets of X with at most n points. Given n≥2, the n-fold symmetric product suspension of X is the quotient space Fn(X)/F1(X) which is obtained from Fn(X) by identifying F1(X) into a one-point set. In this paper we prove that if n∈{2,3}, X is a finite graph, and Y is a continuum such that Fn(X)/F1(X) is homeomorphic to Fn(Y)/F1(Y), then X is homeomorphic to Y. This result completes the work of Germán Montero-Rodríguez, David Herrera-Carrasco, María de J. López and Fernando Macías-Romero who previously proved the corresponding result, for each n≥4.

Uniqueness of the (n,m)-fold hyperspace suspension for continua

Let n,m∈N with m≤n and X be a metric continuum. We consider the hyperspaces Cn(X) (respectively, Fn(X)) of all nonempty closed subsets of X with at most n components (respectively, n points). The (n,m)-fold hyperspace suspension on X was introduced in 2018 by Anaya, Maya, and Vázquez-Juárez, to be the quotient space Cn(X)/Fm(X) which is obtained from Cn(X) by identifying Fm(X) into a one-point set. In this paper: we present some properties of this hyperspace. In particular, if X is a meshed continuum, Y is a continuum and Cn(X)/Fm(X) is homeomorphic to Cn(Y)/Fm(Y), then X is homeomorphic to Y. Furthermore, we prove that if X,Y are almost meshed locally connected continua such that Cn(X)/Fm(X) is homeomorphic to Cn(Y)/Fm(Y), then X is homeomorphic to Y, for each n>2 and m>1. We also provide examples of almost meshed locally connected continua X which do not have unique hyperspace Cn(X)/Fm(X).

Propiedades del (n, m)−ésimo hiperespacio suspensión de continuos

Let n, m ∈ N with m ≤ n and X be a metric continuum. We consider the hyperspaces Cn(X) (respectively, Fn(X)) of all nonempty closed subsets of X with at most n components (respectively, n points). The (n, m)−fold hyperspace suspension on X was introduced in 2018 by Anaya, Maya, and Vázquez-Juárez, to be the quotient space Cn(X)/Fm(X) which is obtained from Cn(X) by identifying Fm(X) into a one-point set. In this paper we prove that Cn(X)/Fm(X) contains an n−cell; Cn(X)/Fm(X) has property (b); Cn(X)/Fm(X) is unicoherent; Cn(X)/Fm(X) is colocally connected; Cn(X)/Fm(X) is aposyndetic; and Cn(X)/Fm(X) is finitely aposyndetic.

Hyperspaces through regular and meager subcontinua

Given a metric continuum X, let C(X) and F1(X) be the hyperspaces of subcontinua and the one-point sets of X, respectively. Let D(X) be the collection of all regular subsets in X belonging to C(X) and let M(X) be all the subcontinua of X with empty interior. In the first part of this paper we are interested in the problem to classify continua that satisfies one of the following equalities: F1(X)=M(X), D(X)=D(X)−F1(X), D(X)=C(X)−M(X) and C(X)=D(X)∪M(X). We show that for hereditarily locally connected continua these equalities and the property of not contain a continuum called dendrite D1 are equivalent each other. In the second part, we consider a continuum X for which there exists a continuous surjective and monotone function f:X→[0,1]. We show that the conditions that X is a λ-type continuum and every t in [0,1] is a cohesion point are equivalent to the equalities D(X)={A∈C(X):f(A)∈D([0,1])} and M(X)={A∈C(X):f(A)∈M([0,1])}. Throughout this paper we pose open problems.

The hyperspace of connected boundary subcontinua of a continuum

Given a metric continumm X, let CB(X) be the hyperspace of subcontinua of X with connected boundary. In this paper we present results concerning CB(X), first about continua for which every subcontinuum has connected boundary. Then we study continua that only have one-point sets as connected boundary proper subcontinua. Later we include characterizations of arcs and simple closed curves. Finally we give a formula for the number of components of CB(X) for finite graphs.

Tautness for sets of multiples and applications to ${\mathcal B}$-free dynamics

For any set $\mathcal B\subseteq\mathbb N=\{1,2,\dots\}$ one can define its \emph{set of multiples} $\mathcal M_{\mathcal B}:=\bigcup_{b\in\mathcal B}b\mathbb Z$ and the set of \emph{$\mathcal B$-free numbers} $\mathcal F_{\mathcal B}:=\mathbb Z\setminus\mathcal M_{\mathcal B}$. Tautness of the set $\mathcal B$ is a basic property related to questions around the asymptotic density of $\mathcal M_{\mathcal B}\subseteq\mathbb Z$. From a dynamical systems point of view (originated by Sarnak) one studies $\eta$, the indicator function of $\mathcal F_{\mathcal B}\subseteq\mathbb Z$, its shift-orbit closure $X_\eta\subseteq\{0,1\}^{\mathbb Z}$ and the stationary probability measure $\nu_\eta$ defined on $X_\eta$ by the frequencies of finite blocks in $\eta$. In this paper we prove that tautness implies the following two properties of $\eta$: (1) The measure $\nu_\eta$ has full topological support in $X_\eta$. (2) If $X_\eta$ is proximal, i.e. if the one-point set $\{\dots000\dots\}$ is contained in $X_\eta$ and is the unique minimal subset of $X_\eta$, then $X_\eta$ is hereditary, i.e. if $x\in X_\eta$ and if $w$ is an arbitrary element of $\{0,1\}^{\mathbb Z}$, then also the coordinate-wise product $w\cdot x$ belongs to $X_\eta$. This strengthens two results from [Bartnicka et al. 2015] which need the stronger assumption that $\mathcal B$ has light tails for the same conclusions.

Uniqueness Theorems for Franklin Series

Simple Franklin series are investigated that converge to zero everywhere except for one point (or several points). It is also proved that the one-point set (or a finite set) is a uniqueness set for Pringsheim convergent double series.

On dependence of meromorphic functions sharing some finite sets IM

In connection with Nevanlinna’s five-value theorem ([2]), the author showed in~[3] that two meromorphic functions sharing five one-point or two-point sets IM are Möbius transforms of each other. Now, we consider $n+1$ meromorphic functions sharing some finite sets IM.

A holonomy invariant anisotropic surface energy in a Riemannian manifold

In this paper, we investigate a holonomy invariant elliptic anisotropic surface energy for hypersurfaces in a complete Riemannian manifold, where “holonomy invariant” means that the elliptic parametric Lagrangian (i.e., a Finsler metric) of the Riemannian manifold used to define the anisotropic surface energy is constant along each holonomy subbundle of the tangent bundle of the Riemannian manifold. First we obtain the first variational formula for this anisotropic surface energy. Next we shall introduce the notions of an anisotropic convex hypersurface, an anisotropic equifocal hypersurface and an anisotropic isoparametric hypersurface for this anisotropic surface energy. Also, we shall introduce the notion of an anisotropic tube for this anisotropic surface energy. We prove that anisotropic tubes over a one-point set in a symmetric space are anisotropic convex hypersurfaces and that anisotropic tubes over a certain kind of reflective submanifold in a symmetric space are anisotropic isoparametric and anisotropic equifocal hypersurfaces.

Framed continua have unique n-fold hyperspace suspension

For a metric continuum X and a positive integer n, we consider the hyperspaces Cn(X) (respectively, Fn(X)) of all nonempty closed subsets of X with at most n components (respectively, n points). Let HSn(X) be the quotient space Cn(X)/Fn(X) which is obtained from Cn(X) by identifying Fn(X) into a one-point set. In this paper we prove that if X is a framed continuum and Y is a continuum such that HSn(X) is homeomorphic to HSn(Y), then X is homeomorphic to Y. This result generalizes the work of David Herrera-Carrasco, Alejandro Illanes, Fernando Macías-Romero and Francisco Vázquez-Juárez who previously proved the corresponding result for finite graphs. This answers a question by Alejandro Illanes.

A note on compact-continuous mappings and k-separation axioms

We study some properties of compact-continuous mappings. We prove that if f:X→Y is a compact-continuous surjection and X is a Lindelöf Σ-space, then Y is a Lindelöf Σ-space. A space X is called a weakly Tychonoff space, if one-point sets are closed in X and for each point x0 and each closed set A not containing x0, there is a compact-continuous function f:X→[0,1] such that f(x0)=1 and f(A)⊂{0}. Some properties of weakly Tychonoff spaces are discussed in this note. We show that if X is a weakly Tychonoff Lindelöf Σ-space and f:X→Y is a compact-continuous bijection such that iw(Y)≤ω, then X has a countable network.Let (X,T) be a topological space. The topological space (X,Tk) is called a k-leader of (X,T), where Tk={U:U∩C is open in C for each compact subspace C of X}. In the last part of this note we get some conclusions on separation axioms of (X,Tk).

Whitney levels in hyperspaces of non-metrizable continua

Let X be a Hausdorff continuum (a nondegenerate, compact, connected, Hausdorff space). Let C(X) (respectively F1(X)) denote the hyperspace of its subcontinua (respectively, its one-point sets), endowed with the Vietoris topology. In this paper we introduce the definition of Whitney levels in C(X) and discuss some basic properties. With this definition, the subsets F1(X) and {X} of C(X) are Whitney levels in C(X), so we call them trivial Whitney levels. In the particular case when X is a generalized arc, we give a condition for the existence of non-trivial Whitney levels in its hyperspace of subcontinua. Finally, we apply this result to the study of Whitney levels in C(X) when X is the Long Arc and the Lexicographic Square.

On meromorphic functions sharing a one-point set and three two-point sets CM

We show that if two meromorphic functions sharing a one-point set and three two-point sets CM, then one of them is a Mobius transform of the other.

On meromorphic functions sharing two one-point sets and two three-point sets

On meromorphic functions sharing two one-point sets and two three-point sets

On meromorphic functions sharing five one-point or two-point sets IM

We show that if two meromorphic functions sharing five one-point or two-point sets two points IM, then one of them is a Mobius transformation of the other.

On meromorphic functions sharing two one-point sets and two two-point sets

We give a uniqueness theorem of two meromorphic functions sharing two one-point sets and two two-point sets CM.

Making holes in hyperspaces

Let X be a metric continuum and C ( X ) the hyperspace of all nonempty subcontinua of X. Let A ∈ C ( X ) , A is said to make a hole in C ( X ) , if C ( X ) − { A } is not unicoherent. In this paper we study the following problem. Problem: For which A ∈ C ( X ) , A makes a hole in C ( X ) . In this paper we present some partial solutions to this problem in the following cases: (1) A is a free arc; (2) A is a one-point set; (3) A is a free simple closed curve; (4) A = X .

Compact operators without extended eigenvalues

A complex number λ is called an extended eigenvalue of a bounded linear operator T on a Banach space B if there exists a non-zero bounded linear operator X acting on B such that X T = λ T X . We show that there are compact quasinilpotent operators on a separable Hilbert space, for which the set of extended eigenvalues is the one-point set { 1 } .