A note on the paper ``Smoothness and the property of Kelley''

Acosta, Gerardo ; Aguilar-Martínez, Álgebra

Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 4, pp. 669-676.

Résumé

Let $X$ be a continuum. In Proposition 31 of J.J. Charatonik and W.J. Charatonik, {\it Smoothness and the property of Kelley\/}, Comment. Math. Univ. Carolin. {\bf 41} (2000), no. 1, 123--132, it is claimed that $L(X) = \bigcap _{p\in X}S(p)$, where $L(X)$ is the set of points at which $X$ is locally connected and, for $p\in X$, $a\in S(p)$ if and only if $X$ is smooth at $p$ with respect to $a$. In this paper we show that such equality is incorrect and that the correct equality is $P(X) = \bigcap _{p\in X}S(p)$, where $P(X)$ is the set of points at which $X$ is connected im kleinen. We also use the correct equality to obtain some results concerning the property of Kelley.

Détail

Classification : 54B20, 54F15, 54F50
Keywords: connectedness im kleinen; continuum; hyperspace; local connectedness; property of Kelley; smoothness