Difference between revisions of "Limit point"
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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 | + | 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>. | ||
{{stub}} | {{stub}} | ||
[[Category:Topology]] | [[Category:Topology]] | ||
Revision as of 09:58, 29 June 2008
Given a topological space 
and a subset 
of , an element 
of is called a limit point of if every neighborhood of contains some element of other than .
When is a metric space, it follows that every neighborhood of must contain infinitely many elements of . A point such that each neighborhood of contains uncountably many elements of is called a condensation point of .
Examples
- Let
be the space of real numbers (with the usual topology) and let 
, that is the set of reciprocals of the positive integers. Then 
is the unique limit point of 
.
be the space of 
, that is the set of 
is the unique limit point of - Let and
be the set of rational numbers. Then every point of 
is a limit point of . Equivalently, we may say that 
is dense in 
.
be the set of 
is 
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