Difference between revisions of "Dense"
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(Created page with "{{Stub page|grade=B|msg=Revise page, add some links to propositions or theorems using the dense property. Also more references, then demote}} ==Definition== Let {{Top.|X|J}} b...") |
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==Definition== | ==Definition== | ||
Let {{Top.|X|J}} be a [[topological space]] and let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset]] of {{M|X}}. We say "''{{M|A}} is dense in {{M|X}}'' if{{rITTMJML}}: | Let {{Top.|X|J}} be a [[topological space]] and let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset]] of {{M|X}}. We say "''{{M|A}} is dense in {{M|X}}'' if{{rITTMJML}}: | ||
| − | * {{M|1=\overline{A}=X}} - that is to say that the {{link|closure|topology}} of {{M|A}} is the entirety of {{M|X}} itself. | + | * {{M|1=\overline{A}=X}} - that is to say that the {{link|closure|set, topology}} of {{M|A}} is the entirety of {{M|X}} itself. |
==See also== | ==See also== | ||
* [[Equivalent statements to a set being dense]] | * [[Equivalent statements to a set being dense]] | ||
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Revise page, add some links to propositions or theorems using the dense property. Also more references, then demote
Definition
Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath]A\in\mathcal{P}(X)[/ilmath] be an arbitrary subset of [ilmath]X[/ilmath]. We say "[ilmath]A[/ilmath] is dense in [ilmath]X[/ilmath] if[1]:
- [ilmath]\overline{A}=X[/ilmath] - that is to say that the closure of [ilmath]A[/ilmath] is the entirety of [ilmath]X[/ilmath] itself.
