Difference between revisions of "Dense set"
From Encyclopedia of Mathematics
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| − | A subset $A$ of topological space $X$ is dense | + | A subset $A$ of a [[topological space]] $X$ is dense for which the [[Closure of a set|closure]] is the entire space $X$ (some authors use the terminology ''everywhere dense''). A common alternative definition is: |
* a set $A$ which intersects every nonempty open subset of $X$. | * a set $A$ which intersects every nonempty open subset of $X$. | ||
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A set which is not dense in any non-empty open subset of a topological space $X$ is called [[Nowhere-dense set|nowhere dense]]. | A set which is not dense in any non-empty open subset of a topological space $X$ is called [[Nowhere-dense set|nowhere dense]]. | ||
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| + | A set which consists of limit points is called [[Dense-in-itself set|dense-in-itself]]. | ||
Latest revision as of 20:33, 13 December 2017
2020 Mathematics Subject Classification: Primary: 54A05 [MSN][ZBL]
A subset $A$ of a topological space $X$ is dense for which the closure is the entire space $X$ (some authors use the terminology everywhere dense). A common alternative definition is:
- a set $A$ which intersects every nonempty open subset of $X$.
If $U\subset X$, a set $A\subset X$ is called dense in $U$ if $A\cap U$ is a dense set in the subspace topology of $U$. When $U$ is open this is equivalent to the requirement that the closure (in $X$) of $A$ contains $U$.
A set which is not dense in any non-empty open subset of a topological space $X$ is called nowhere dense.
A set which consists of limit points is called dense-in-itself.
How to Cite This Entry:
Dense set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dense_set&oldid=28109
Dense set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dense_set&oldid=28109
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article