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| − | ''at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s0858601.png" />'' | + | {{TEX|done}} |
| | + | ''at a point $p$'' |
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| − | A [[Continuum|continuum]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s0858602.png" /> such that for each sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s0858603.png" /> of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s0858604.png" /> converging towards a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s0858605.png" /> and each subcontinuum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s0858606.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s0858607.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s0858608.png" /> there exist a sequence of subcontinua <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s0858609.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s08586010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s08586011.png" />, converging towards <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s08586012.png" />. A continuum that is smooth at each one of its points is called smooth. | + | A [[Continuum|continuum]] $X$ such that for each sequence $x_1,\ldots,x_n,\ldots,$ of points of $X$ converging towards a point $x$ and each subcontinuum $K\subset X$ containing $p$ and $x$ there exist a sequence of subcontinua $\{K_n\}$ in $X$, $p,x_n\in K_n$, converging towards $K$. A continuum that is smooth at each one of its points is called smooth. |
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| | ====Comments==== | | ====Comments==== |
| − | Smoothness has a slightly different definition in the class of uniquely arcwise-connected continua, or dendroids (a continuum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s08586013.png" /> is uniquely arcwise-connected if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s08586014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s08586015.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s08586016.png" /> there is a unique arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s08586017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s08586018.png" /> connecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s08586019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s08586020.png" />). One calls a uniquely arcwise-connected continuum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s08586021.png" /> smooth if there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s08586022.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s08586023.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s08586024.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s08586025.png" />. Such a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s08586026.png" /> is called an initial point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085860/s08586027.png" />. | + | Smoothness has a slightly different definition in the class of uniquely arcwise-connected continua, or dendroids (a continuum $X$ is uniquely arcwise-connected if for every $x$ and $y$ in $X$ there is a unique arc $[x,y]$ in $X$ connecting $x$ and $y$). One calls a uniquely arcwise-connected continuum $X$ smooth if there is a $p\in X$ such that $[p,x_n]\to[p,x]$ whenever $x_n\to x$ in $X$. Such a point $p$ is called an initial point of $X$. |
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| | ====References==== | | ====References==== |
| | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.J. Charatonik, C. Eberhart, "On smooth dendroids" ''Fund. Math.'' , '''67''' (1970) pp. 297–322</TD></TR></table> | | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.J. Charatonik, C. Eberhart, "On smooth dendroids" ''Fund. Math.'' , '''67''' (1970) pp. 297–322</TD></TR></table> |
Revision as of 13:47, 29 April 2014
at a point $p$
A continuum $X$ such that for each sequence $x_1,\ldots,x_n,\ldots,$ of points of $X$ converging towards a point $x$ and each subcontinuum $K\subset X$ containing $p$ and $x$ there exist a sequence of subcontinua $\{K_n\}$ in $X$, $p,x_n\in K_n$, converging towards $K$. A continuum that is smooth at each one of its points is called smooth.
Smoothness has a slightly different definition in the class of uniquely arcwise-connected continua, or dendroids (a continuum $X$ is uniquely arcwise-connected if for every $x$ and $y$ in $X$ there is a unique arc $[x,y]$ in $X$ connecting $x$ and $y$). One calls a uniquely arcwise-connected continuum $X$ smooth if there is a $p\in X$ such that $[p,x_n]\to[p,x]$ whenever $x_n\to x$ in $X$. Such a point $p$ is called an initial point of $X$.
References
| [a1] | J.J. Charatonik, C. Eberhart, "On smooth dendroids" Fund. Math. , 67 (1970) pp. 297–322 |
How to Cite This Entry:
Smooth continuum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Smooth_continuum&oldid=11868
This article was adapted from an original article by A.A. Mal'tsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article