arXiv subjects
Publications and source records attributed to Verónica Martínez-de-la-Vega.
Given a continuum $X$, let $C(X)$ be the hyperspace of all subcontinua of $X$. We consider the hyperspace $NC^{*}(X)=\{A\in C(X):X\setminus A$ is connected$\}$. In this paper we prove that the only locally connected continua $X$ for which $NC^{*}(X)$ is compact are the arcs and the simple closed curves. We also characterize the finite graphs $G$ for which $NC^{*}(G)$ is connected.
Given a continuum $X$, let $C(X)$ denote the hyperspace of all subcontinua of $X$. In this paper we study the Vietoris hyperspace $NC^{*}(X)=\{ A \in C(X):X\setminus A\text{ is connected}\}$ when $X$ is a finite graph or a dendrite; in particular, we give conditions under which $NC^{*}(X)$ is compact, connected, locally connected or totally disconnected. Also, we prove that if $X$ is a dendrite and the set of endpoints of $X$ is dense, then $NC^{*}(X)$ is homeomorphic to the Baire space of irrational numbers.
A metric continuum $X$ is indecomposable if it cannot be put as the union of two of its proper subcontinua. A subset $R$ of $X$ is said to be continuumwise connected provided that for each pair of points $p,q\in R$, there exists a subcontinuum $M$ of $X$ such that $\{p,q\}\subset M\subset R$. Let $X^{2}$ denote the Cartesian square of $X$ and $Δ$ the diagonal of $X^{2}$. In \cite{ka} it was asked if for a continuum $X$, distinct from the arc, $X^{2}\setminus Δ$ is continuumwise connected if and only if $X$ is decomposable. In this paper we show that no implication in this question holds. For the proof of the non-necessity, we use the dynamic properties of a suitable homeomorphism of the Cantor set onto itself to construct an appropriate indecomposable continuum $X$.
For a given continuum $X$ and a natural number $n,$ we consider the hyperspace $F_n(X)$ of all nonempty subsets of $X$ with at most $n$ points, metrized by the Hausdorff metric. In this paper we show that if $X$ is a dendrite whose set of end points is closed, $n \in \mathbb{N}$ and $Y$ is a continuum such that the hyperspaces $F_n(X)$ and $F_n(Y)$ are homeomorphic, then $Y$ is a dendrite whose set of end points is closed.
For a metric continuum $X$, we consider the $n^{\tiny\textrm{th}}$-symmetric product $F_{n}(X)$ defined as the hyperspace of all nonempty subsets of $X$ with at most $n$ points. The homogeneity degree $hd(X)$ of a continuum $X$ is the number of orbits for the action of the group of homeomorphisms of $X$ onto itself. In this paper we determine $hd(F_{n}(X))$ for every manifold without boundary and $n\in \mathbb{N}$. We also compute $hd(F_{n}[0,1])$ for all $n\in \mathbb{N}$.