On Jones' set function \(\mathcal{T}\) and the property of Kelley for Hausdorff continua
The set function \(\mathcal{T}\) was defined by \textit{F. B. Jones} in [Am. J. Math. 70, 403--413 (1948; Zbl 0035.10904)] for metrizable continua. It was first looked at in the class of Hausdorff continua in [\textit{H. S. Davis} et al., Port. Math. 21, 113--133 (1962; Zbl 0107.40101)]. In the following, a continuum is a compact connected Hausdorff space. Given a continuum \(X\), we define the set function \(\mathcal{T}\) as follows: if \(A\subset X\), then \(\mathcal{T}(A)=X\setminus\{x\in X\,|\,\mathrm{there\,\,exists\,\,a\,subcontinuum} \break W\,\mathrm{of}\,X\,\mathrm{such\,that}\,x\in\,\mathrm{Int}(W)\subset X\setminus A\}\). The author proves a theorem that extends certain work of Camargo and Uzcátegui from metrizable continua. \smallskip Theorem 3.4. Let \(X\) be a continuum. If \(\mathcal{T}\) is continuous and idempotent on singletons, then \(\{\mathcal{T}(\{x\})\,|\,x\in X\}\) is a decomposition of \(X\). \smallskip In Section 4, the author goes on to study continua with the property of Kelley and how \(\mathcal{T}\) operates within that class. In Section 5 (Theorem 5.6), a weak version of Jones' Aposyndetic Decomposition Theorem for homogeneous continua having the property of Kelley is obtained. A ``full'' version of this (Theorem 5.13) is obtained under what is called the uniform property of Effros. \smallskip Theorem 5.13. Let \(X\) be a homogeneous continuum with the uniform property of Effros and \(\mathcal{G}=\{\mathcal{T}(\{x\})\,|\, x\in X\}\). Then \(\mathcal{G}\) is a continuous terminal decomposition of \(X\), \(X/\mathcal{G}\) is an aposyndetic homogeneous continuum, and the quotient map \(q:X\to\mathcal{G}\) is an atomic map.
scientific article; zbMATH DE number 6728655
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