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

Acosta, Gerardo, and Aguilar-Martínez, Álgebra. "A note on the paper ``Smoothness and the property of Kelley''." Commentationes Mathematicae Universitatis Carolinae 48.4 (2007): 669-676. <http://eudml.org/doc/250228>.

@article{Acosta2007,
abstract = {Let $X$ be a continuum. In Proposition 31 of J.J. Charatonik and W.J. Charatonik, Smoothness and the property of Kelley, Comment. Math. Univ. Carolin. 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.},
author = {Acosta, Gerardo, Aguilar-Martínez, Álgebra},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {connectedness im kleinen; continuum; hyperspace; local connectedness; property of Kelley; smoothness; continuum; connectedness im kleinen; local connectedness; property of Kelley; smoothness},
language = {eng},
number = {4},
pages = {669-676},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A note on the paper ``Smoothness and the property of Kelley''},
url = {http://eudml.org/doc/250228},
volume = {48},
year = {2007},
}

TY - JOUR
AU - Acosta, Gerardo
AU - Aguilar-Martínez, Álgebra
TI - A note on the paper ``Smoothness and the property of Kelley''
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 4
SP - 669
EP - 676
AB - Let $X$ be a continuum. In Proposition 31 of J.J. Charatonik and W.J. Charatonik, Smoothness and the property of Kelley, Comment. Math. Univ. Carolin. 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.
LA - eng
KW - connectedness im kleinen; continuum; hyperspace; local connectedness; property of Kelley; smoothness; continuum; connectedness im kleinen; local connectedness; property of Kelley; smoothness
UR - http://eudml.org/doc/250228
ER -