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Kelley remainders of \([0,\infty)\). (English) Zbl 1161.54014

A metric continuum \(X\) is called a Kelley continuum provided that for each point \(p\), each continuum \(A\) in \(X\) containing \(p\) and each sequence of \(p_n\) converging to \(p\), there exists a sequence of subcontinua \(A_n\) containing \(p_n\) that converges to \(A\).
Even though Kelley continua were introduced to study contractibility in hyperspaces, they have become interesting in their own right and have been studied extensively.
In the paper under review, the authors investigate Kelley remainders, that is, Kelley continua that arise as the remainders of Kelley compactifications of the ray \([0,1)\). The following results are obtained: (a) Chainable Kelley continua are Kelley remainders; (b) Kelley continua in which each nondegenerate proper subcontinuum is an arc are Kelley remainders; (c) There exists exactly one (up to homeomorphisms) compactification \(X\) of \([0,1)\) with remainder a simple closed curve such that \(X\) is a Kelley continuum (the corresponding result for the arc was known, see Theorem 16.28 of [S. B. Nadler jun., Hyperspaces of sets. A text with research questions. Monographs and Textbooks in Pure and Applied Mathematics. Vol. 49. New York-Basel: Marcel Dekker, Inc. (1978; Zbl 0432.54007)]); (d) The property of being a Kelley remainder is not hereditary; (e) The property of being a Kelley remainder is preserved under confluent maps. The authors include the following three open questions: (a) How can Kelley reaminders be characterized? (b) Is every Kelley circle-like continuum a Kelley remainder? (c) Is every atriodic, tree-like Kelley continuum a Kelley remainder?

MSC:

54F15 Continua and generalizations
54F50 Topological spaces of dimension \(\leq 1\); curves, dendrites
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54D40 Remainders in general topology

Citations:

Zbl 0432.54007
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