The property of Kelley by arcs

期刊论文
出版模式：

开放获取
发表日期：

2024年8月8日

查看全文

Mauricio Chacón-Tirado,
María de J. López,
José Luis Suárez-López

摘要

For a metric continuum X and a point $p\in X$ $p \in X$ , the hyperspace of arcs in X containing p, Arcs(p, X), is defined as the set containing $\{p\}$ ${p}$ and all arcs in X that contain the point p, we endow this hyperspace with the Hausdorff metric. In this paper we introduce the property of Kelley by arcs; a continuum X has the property of Kelley by arcs provided that for each $p\in X$ $p \in X$ , for each $A\in Arcs(p, X)$ $A \in A r c s (p, X)$ , and for each sequence of points $\{p_{n}\}_{n\in \mathbb {N}}$ ${p_{n}}_{n \in N}$ that converges to p, there exists a sequence $\{A_{n}\}_{n\in \mathbb {N}}$ ${A_{n}}_{n \in N}$ that converges to A such that $A_{n}\in Arcs(p_{n}, X)$ $A_{n} \in A r c s (p_{n}, X)$ , for each $n\in \mathbb {N}$ $n \in N$ . We show that in the class of finite graphs the property of Kelley by arcs characterizes the fact of being either an arc or a simple closed curve. We prove that in arc-continua the property of Kelley by arcs is equivalent to the property of Kelley. Also, we prove that homogeneous continua have the property of Kelley by arcs. Moreover, we prove that in the class of dendroids the property of Kelley by arcs implies the property of Kelley, and we show that the arc is the only dendroid with the property of Kelley by arcs. Finally, we study the property of Kelley by arcs in the class of compactifications of the ray $[0, \infty )$ $[0, \infty)$ with remainder a finite graph or a dendroid.