Article

 Title: A note on the paper Smoothness and the property of Kelley''  (English) Author: Acosta, Gerardo Author: Aguilar-Martínez, Álgebra Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 48 Issue: 4 Year: 2007 Pages: 669-676 . Category: math . Summary: 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. Keyword: connectedness im kleinen Keyword: continuum Keyword: hyperspace Keyword: local connectedness Keyword: property of Kelley Keyword: smoothness MSC: 54B20 MSC: 54F15 MSC: 54F50 idZBL: Zbl 1199.54183 idMR: MR2375167 . Date available: 2009-05-05T17:05:32Z Last updated: 2012-05-01 Stable URL: http://hdl.handle.net/10338.dmlcz/119689 . Reference: [1] Acosta G.: On smooth fans and unique hyperspace.Houston J. Math. 30 (2004), 99-115. MR 2048337 Reference: [2] Acosta G., Illanes A.: Continua which have the property of Kelley hereditarily.Topology Appl. 102 (2000), 151-162. Zbl 0940.54038, MR 1741483 Reference: [3] Charatonik J.J., Charatonik W.J.: Smoothness and the property of Kelley.Comment Math. Univ. Carolin. 41 1 (2000), 123-132. Zbl 1037.54506, MR 1756932 Reference: [4] Maćkowiak T.: On smooth continua.Fund. Math. 85 (1974), 79-95. MR 0365532 Reference: [5] Nadler S.B., Jr.: Hyperspaces of Sets.Marcel Dekker, Inc., New York and Basel, 1978. Zbl 1125.54001, MR 0500811 Reference: [6] Nadler S.B., Jr.: Continuum Theory.Marcel Dekker, Inc., New York, Basel and Hong Kong, 1992. Zbl 0819.54015, MR 1192552 .

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