Difference between revisions of "Limit point"

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Let <math>X</math> be a [[topological space]]; let <math>S</math> be a subset of <math>X</math>.  An element <math>x</math> of <math>X</math> is called a '''limit point''' of <math>S</math> if every [[neighborhood]] of <math>x</math> contains some element of <math>S</math> other than <math>x</math>.
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Given a [[topological space]] <math>X</math> and a [[subset]] <math>S</math> of <math>X</math>, an [[element]] <math>x</math> of <math>X</math> is called a '''limit point''' of <math>S</math> if every [[neighborhood]] of <math>x</math> contains some element of <math>S</math> other than <math>x</math>.
  
When <math>X</math> is a [[metric space]], it follows that every neighborhood of <math>x</math> must contain infinitely many elements of <math>S</math> other than <math>x</math>.  A point <math>x</math> such that each neighborhood of <math>x</math> contains [[uncountably many]] elements of <math>S</math> is called a '''condensation point''' of <math>S</math>.
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When <math>X</math> is a [[metric space]], it follows that every neighborhood of <math>x</math> must contain [[infinite]]ly many elements of <math>S</math>.  A point <math>x</math> such that each neighborhood of <math>x</math> contains [[uncountably many]] elements of <math>S</math> is called a '''condensation point''' of <math>S</math>.
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== Examples ==
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* Let <math>X = \mathbb{R}</math> be the space of [[real number]]s (with the usual topology) and let <math>S = \{\frac{1}{n} \mid n \in \mathbb{Z}_{> 0}\}</math>, that is the set of [[reciprocal]]s of the [[positive integer]]s.  Then <math>0</math> is the unique limit point of <math>S</math>.
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* Let <math>X = \mathbb{R}</math> and <math>S =\mathbb{Q}</math> be the set of [[rational number]]s.  Then every point of <math>X</math> is a limit point of <math>S</math>.  Equivalently, we may say that <math>\mathbb{Q}</math> is [[dense]] in <math>\mathbb{R}</math>.
  
 
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[[Category:Analysis]]
 
[[Category:Topology]]
 
[[Category:Topology]]

Latest revision as of 17:50, 28 March 2009

Given a topological space $X$ and a subset $S$ of $X$, an element $x$ of $X$ is called a limit point of $S$ if every neighborhood of $x$ contains some element of $S$ other than $x$.

When $X$ is a metric space, it follows that every neighborhood of $x$ must contain infinitely many elements of $S$. A point $x$ such that each neighborhood of $x$ contains uncountably many elements of $S$ is called a condensation point of $S$.

Examples

  • Let $X = \mathbb{R}$ and $S =\mathbb{Q}$ be the set of rational numbers. Then every point of $X$ is a limit point of $S$. Equivalently, we may say that $\mathbb{Q}$ is dense in $\mathbb{R}$.

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