Continua which have the property of Kelley hereditarily (Q1971122)

A metric continuum \(X\) with a metric \(d\) is said to have the property of Kelley provided that for each point \(x\in X\) and for each \(\varepsilon> 0\) there is a \(\delta>0\) such that if a point \(b\in X\) and a continuum \(A\subset X\) satisfy \(d(a,b)< \delta\) and \(a\in A\), then there exists a continuum \(B\subset X\) containing \(b\) which is \(\varepsilon\)-near to \(A\). A continuum \(X\) is said to have the property of Kelley hereditarily (concisely pKh) provided that each of the subcontinua of \(X\) has the property of Kelley. The authors investigate continua having pKh. Among other results it is shown that a continuum \(X\) is hereditarily locally connected if and only if \(X\) has pKh and is arcwise connected. This generalizes a result of \textit{S. T. Czuba} [Proc. Am. Math. Soc. 102, No. 3, 728-730 (1988; Zbl 0648.54030)]. The paper consists of 10 sections: 1. Introduction. 2. The property of Kelly and cut points. 3. Composants. 4. Hereditary local connectedness and the property of Kelley hereditarily. 5. The property of Kelley and \(\infty\)-ods. 6. Compactifications of \([0,\infty)\) with the property of Kelley. 7. Mappings. 8. Hyperspaces and products. 9. Whitney properties. 10. Homogeneous continua and the property of Kelley hereditarily.