Tsukuba Journal of Mathematics

Zero-dimensional subsets of hyperspaces

Alejandro Illanes

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Abstract

Let $X$ be a metric continuum, let $2^{X}$ be the hyperspaces of all the nonempty closed subsets of $X$ and let $C(X)$ be the hyperspace of subcontinua of $X$. In this paper we prove: THEOREM 1. If $\mathscr{H}$ is a O-dimensional subset of $2^{X}$, then $2^{X}-\mathscr{H}$ is connected. THEOREM 2. If $\mathscr{H}$ is a closed O-dimensional subset of $C(X)$ such that $C(X)-\{A\}$ is arcwise connected for each $A\in \mathscr{H}$,then $C(X)-\mathscr{H}$ is arcwise connected. Theorem 2 answers a question by Sam B. Nadler, Jr.

Article information

Source
Tsukuba J. Math., Volume 24, Number 2 (2000), 249-255.

Dates
First available in Project Euclid: 30 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.tkbjm/1496164148

Digital Object Identifier
doi:10.21099/tkbjm/1496164148

Mathematical Reviews number (MathSciNet)
MR1818085

Zentralblatt MATH identifier
0980.54007

Citation

Illanes, Alejandro. Zero-dimensional subsets of hyperspaces. Tsukuba J. Math. 24 (2000), no. 2, 249--255. doi:10.21099/tkbjm/1496164148. https://projecteuclid.org/euclid.tkbjm/1496164148


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