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| In [[mathematics]], a [[subset]] <math>A</math> of a [[topological space]] is said to be '''dense-in-itself''' if <math>A</math> contains no [[isolated point]]s.
| | The author's title is Christy Brookins. For years he's been residing in Alaska and he doesn't strategy on altering it. My working day occupation is a travel agent. To play lacross is the factor I adore most of all.<br><br>My weblog - real psychics ([http://kard.dk/?p=24252 kard.dk]) |
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| Every dense-in-itself [[closed set]] is [[perfect set|perfect]]. Conversely, every perfect set is dense-in-itself.
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| A simple example of a set which is dense-in-itself but not closed (and hence not a perfect set) is the subset of [[irrational numbers]] (considered as a subset of the [[real numbers]]). This set is dense-in-itself because every [[Neighbourhood (mathematics)|neighborhood]] of an irrational number <math>x</math> contains at least one other irrational number <math>y \neq x</math>. On the other hand, this set of irrationals is not closed because every [[rational number]] lies in its [[closure (topology)|closure]]. For similar reasons, the set of rational numbers (also considered as a subset of the [[real numbers]]) is also dense-in-itself but not closed.
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| The above examples, the irrationals and the rationals, are also [[dense set]]s in their topological space, namely <math>\mathbb{R}</math>. As an example that is dense-in-itself but not dense in its topological space, consider <math>\mathbb{Q} \cap [0,1]</math>. This set is not dense in <math>\mathbb{R}</math> but is dense-in-itself.
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| It is also interesting to note, although tautological, that the domain of a [[continuous function]] must be the union of dense-in-itself sets and/or isolated points.{{fact|date=January 2013}}
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| ==See also==
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| * [[Nowhere dense set]]
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| * [[Dense order]]
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| * [[Perfect space]]
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| * [[Glossary of topology]]
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| ==References==
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| {{reflist}}
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| *{{cite book | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | origyear=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | page=6 }}
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| {{PlanetMath attribution|id=6228|title=Dense in-itself}}
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| [[Category:Topology]]
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| {{topology-stub}}
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Latest revision as of 01:22, 18 November 2014
The author's title is Christy Brookins. For years he's been residing in Alaska and he doesn't strategy on altering it. My working day occupation is a travel agent. To play lacross is the factor I adore most of all.
My weblog - real psychics (kard.dk)