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Difference between revisions of "Condensation point"

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See [[Condensation point of a set|Condensation point of a set]]; [[Limit point of a set|Limit point of a set]]; [[Accumulation point|Accumulation point]].
 
See [[Condensation point of a set|Condensation point of a set]]; [[Limit point of a set|Limit point of a set]]; [[Accumulation point|Accumulation point]].
  
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====Comments====
 
====Comments====
The three notions mentioned above should be clearly distinguished. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c0244301.png" /> is a subset of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c0244302.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c0244303.png" /> is a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c0244304.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c0244305.png" /> is an accumulation point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c0244306.png" /> if and only if every neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c0244307.png" /> intersects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c0244308.png" />. It is a condensation point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c0244309.png" /> if and only if every neighbourhood of it contains uncountably many points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c02443010.png" />.
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The three notions mentioned above should be clearly distinguished. If $A$ is a subset of a topological space $X$ and $x$ is a point of $X$, then $x$ is an accumulation point of $A$ if and only if every neighbourhood of $x$ intersects $A\setminus\{x\}$. It is a condensation point of $A$ if and only if every neighbourhood of it contains uncountably many points of $A$.
  
The term limit point is slightly ambiguous. One might call <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c02443011.png" /> a limit point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c02443012.png" /> if every neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c02443013.png" /> contains infinitely many points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c02443014.png" />, but this is not standard. Sometimes one calls <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c02443015.png" /> a limit point of a net (cf. [[Net (of sets in a topological space)|Net (of sets in a topological space)]]) if some subset of this net converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c02443016.png" />. However, most people call <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c02443017.png" /> a cluster point in this case.
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The term limit point is slightly ambiguous. One might call $x$ a limit point of $A$ if every neighbourhood of $x$ contains infinitely many points of $A$, but this is not standard. Sometimes one calls $x$ a limit point of a net (cf. [[Net (of sets in a topological space)|Net (of sets in a topological space)]]) if some subset of this net converges to $x$. However, most people call $x$ a cluster point in this case.
  
In the case of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c02443018.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c02443019.png" /> the notions of a limit point of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c02443020.png" /> and an accumulation point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024430/c02443021.png" /> coincide, and one uses  "accumulation point" .
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In the case of a $T_1$-space $X$ the notions of a limit point of a set $A\subset X$ and an accumulation point of $A$ coincide, and one uses  "accumulation point" .

Latest revision as of 14:15, 19 April 2014

See Condensation point of a set; Limit point of a set; Accumulation point.


Comments

The three notions mentioned above should be clearly distinguished. If $A$ is a subset of a topological space $X$ and $x$ is a point of $X$, then $x$ is an accumulation point of $A$ if and only if every neighbourhood of $x$ intersects $A\setminus\{x\}$. It is a condensation point of $A$ if and only if every neighbourhood of it contains uncountably many points of $A$.

The term limit point is slightly ambiguous. One might call $x$ a limit point of $A$ if every neighbourhood of $x$ contains infinitely many points of $A$, but this is not standard. Sometimes one calls $x$ a limit point of a net (cf. Net (of sets in a topological space)) if some subset of this net converges to $x$. However, most people call $x$ a cluster point in this case.

In the case of a $T_1$-space $X$ the notions of a limit point of a set $A\subset X$ and an accumulation point of $A$ coincide, and one uses "accumulation point" .

How to Cite This Entry:
Condensation point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Condensation_point&oldid=17091
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article