Difference between revisions of "Relatively open"
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Given a [[Subspace topology|subspace]] {{M|Y\subset X}} of a [[Topological space|topological space]] {{M|(X,\mathcal{J})}}, the open sets of {{M|(Y,\mathcal{J}_\text{subspace})}} are said to be '''relatively open'''<ref>Introduction to topology - Third Edition - Mendelson</ref> in {{M|X}} | Given a [[Subspace topology|subspace]] {{M|Y\subset X}} of a [[Topological space|topological space]] {{M|(X,\mathcal{J})}}, the open sets of {{M|(Y,\mathcal{J}_\text{subspace})}} are said to be '''relatively open'''<ref>Introduction to topology - Third Edition - Mendelson</ref> in {{M|X}} | ||
| − | + | Alternatively we may say given a {{M|A\subseteq X}} the family of sets: | |
* {{M|1=\{U_A\vert U_A=A\cap U\text{ for some }U\in\mathcal{J}\} }} | * {{M|1=\{U_A\vert U_A=A\cap U\text{ for some }U\in\mathcal{J}\} }} | ||
| − | are all relatively open | + | are all ''relatively open in {{M|A}}'' |
==See also== | ==See also== | ||
* [[Open set]] | * [[Open set]] | ||
| + | * [[Relatively closed]] | ||
==References== | ==References== | ||
Latest revision as of 18:42, 19 April 2015
Definition
Given a subspace [ilmath]Y\subset X[/ilmath] of a topological space [ilmath](X,\mathcal{J})[/ilmath], the open sets of [ilmath](Y,\mathcal{J}_\text{subspace})[/ilmath] are said to be relatively open[1] in [ilmath]X[/ilmath]
Alternatively we may say given a [ilmath]A\subseteq X[/ilmath] the family of sets:
- [ilmath]\{U_A\vert U_A=A\cap U\text{ for some }U\in\mathcal{J}\}[/ilmath]
are all relatively open in [ilmath]A[/ilmath]
See also
References
- ↑ Introduction to topology - Third Edition - Mendelson
