On the property of Kelley in the hyperspace and Whitney continua (Q1114185)

A (metric) continuum X is said to have property K if for each subcontinuum A of X, each point a in A and for each positive \(\epsilon\) there exists a positive \(\delta\) such that if b is a point of X at the distance \(<\delta\) from a then there exists a subcontinuum B of X containing b and being at the Hausdorff distance \(<\epsilon\) from A. This property was discovered by \textit{J. L. Kelley} [Trans. Am. Math. Soc. 52, 22-36 (1942; Zbl 0061.401)] who proved that the hyperspaces C(X) and \(2^ X\) are contractible if X has this property. Later this property was studied as interesting in itself by \textit{R. W. Wardle} [Houston J. Math. 3, 291-299 (1977; Zbl 0355.54025); see also book by \textit{B. S. Nadler jun.}, ``Hyperspaces of sets'' (1978; Zbl 0432.54007)]. The author introduces a much more restrictive property \(K^*\) so that a two-spiral curve from the Wardle paper, having obviously the property K, has no longer the property \(K^*\). However, locally connected continua, homogeneous continua and \(\sin(1/x)\) curve, which are known as having the property K, have the property \(K^*\). The author proves that if X has property \(K^*\) then C(X) has property \(K^*\), and therefore the property K, too. This is the main result of the paper (it is a still open question whether C(X) should have property K if X has property K). Another result: the property \(K^*\) is preserved under countable products. It should be mentioned that the property K, as was shown by Wardle, is not necessarily preserved even under operation of squaring the space.