Is it possible to represent an open and connected set in $(mathbb{R}^n,||cdot||)$ as the finite union of...












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general-topology metric-spaces connectedness







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    It seems doubtful in general. For example, I doubt you could represent the exterior of the unit ball in that way.
    $endgroup$
    – MPW
    Dec 7 '18 at 19:08
















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$begingroup$


A similar question was asked here, but there was no setting given.






general-topology metric-spaces connectedness







share|cite|improve this question











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  • 2




    $begingroup$
    It seems doubtful in general. For example, I doubt you could represent the exterior of the unit ball in that way.
    $endgroup$
    – MPW
    Dec 7 '18 at 19:08














0












0








0





$begingroup$


A similar question was asked here, but there was no setting given.






general-topology metric-spaces connectedness







share|cite|improve this question











$endgroup$




A similar question was asked here, but there was no setting given.






general-topology metric-spaces connectedness




general-topology metric-spaces connectedness






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share|cite|improve this question













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edited Dec 7 '18 at 19:12









José Carlos Santos

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157k22126227










asked Dec 7 '18 at 19:06









LenCLenC

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  • 2




    $begingroup$
    It seems doubtful in general. For example, I doubt you could represent the exterior of the unit ball in that way.
    $endgroup$
    – MPW
    Dec 7 '18 at 19:08














  • 2




    $begingroup$
    It seems doubtful in general. For example, I doubt you could represent the exterior of the unit ball in that way.
    $endgroup$
    – MPW
    Dec 7 '18 at 19:08








2




2




$begingroup$
It seems doubtful in general. For example, I doubt you could represent the exterior of the unit ball in that way.
$endgroup$
– MPW
Dec 7 '18 at 19:08




$begingroup$
It seems doubtful in general. For example, I doubt you could represent the exterior of the unit ball in that way.
$endgroup$
– MPW
Dec 7 '18 at 19:08










1 Answer
1






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No. Take, for instance$$mathbb{R}^2setminusbigcup_{ninmathbb{Z}}bigl({n}times[0,infty)bigr).$$






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  • $begingroup$
    Why is this connected? Its a $xy$-plane separated by vertical lines $x = n$.
    $endgroup$
    – GNUSupporter 8964民主女神 地下教會
    Dec 7 '18 at 19:14










  • $begingroup$
    They're not lines. They're half-lines. It's the plane minus the vertical half-lines starting on points of the type $(n,0)$ and going up from there.
    $endgroup$
    – José Carlos Santos
    Dec 7 '18 at 19:18










  • $begingroup$
    And the reason it can't be written as a finite union of convex sets would be, for example, that each point $(n+frac{1}{2}, 1)$ would have to be in a different one of the convex sets?
    $endgroup$
    – Daniel Schepler
    Dec 7 '18 at 22:04










  • $begingroup$
    @DanielSchepler Yes, that's it.
    $endgroup$
    – José Carlos Santos
    Dec 7 '18 at 22:23











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1 Answer
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1 Answer
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active

oldest

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active

oldest

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active

oldest

votes









3












$begingroup$

No. Take, for instance$$mathbb{R}^2setminusbigcup_{ninmathbb{Z}}bigl({n}times[0,infty)bigr).$$






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Why is this connected? Its a $xy$-plane separated by vertical lines $x = n$.
    $endgroup$
    – GNUSupporter 8964民主女神 地下教會
    Dec 7 '18 at 19:14










  • $begingroup$
    They're not lines. They're half-lines. It's the plane minus the vertical half-lines starting on points of the type $(n,0)$ and going up from there.
    $endgroup$
    – José Carlos Santos
    Dec 7 '18 at 19:18










  • $begingroup$
    And the reason it can't be written as a finite union of convex sets would be, for example, that each point $(n+frac{1}{2}, 1)$ would have to be in a different one of the convex sets?
    $endgroup$
    – Daniel Schepler
    Dec 7 '18 at 22:04










  • $begingroup$
    @DanielSchepler Yes, that's it.
    $endgroup$
    – José Carlos Santos
    Dec 7 '18 at 22:23
















3












$begingroup$

No. Take, for instance$$mathbb{R}^2setminusbigcup_{ninmathbb{Z}}bigl({n}times[0,infty)bigr).$$






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Why is this connected? Its a $xy$-plane separated by vertical lines $x = n$.
    $endgroup$
    – GNUSupporter 8964民主女神 地下教會
    Dec 7 '18 at 19:14










  • $begingroup$
    They're not lines. They're half-lines. It's the plane minus the vertical half-lines starting on points of the type $(n,0)$ and going up from there.
    $endgroup$
    – José Carlos Santos
    Dec 7 '18 at 19:18










  • $begingroup$
    And the reason it can't be written as a finite union of convex sets would be, for example, that each point $(n+frac{1}{2}, 1)$ would have to be in a different one of the convex sets?
    $endgroup$
    – Daniel Schepler
    Dec 7 '18 at 22:04










  • $begingroup$
    @DanielSchepler Yes, that's it.
    $endgroup$
    – José Carlos Santos
    Dec 7 '18 at 22:23














3












3








3





$begingroup$

No. Take, for instance$$mathbb{R}^2setminusbigcup_{ninmathbb{Z}}bigl({n}times[0,infty)bigr).$$






share|cite|improve this answer











$endgroup$



No. Take, for instance$$mathbb{R}^2setminusbigcup_{ninmathbb{Z}}bigl({n}times[0,infty)bigr).$$







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 7 '18 at 21:51

























answered Dec 7 '18 at 19:11









José Carlos SantosJosé Carlos Santos

157k22126227




157k22126227












  • $begingroup$
    Why is this connected? Its a $xy$-plane separated by vertical lines $x = n$.
    $endgroup$
    – GNUSupporter 8964民主女神 地下教會
    Dec 7 '18 at 19:14










  • $begingroup$
    They're not lines. They're half-lines. It's the plane minus the vertical half-lines starting on points of the type $(n,0)$ and going up from there.
    $endgroup$
    – José Carlos Santos
    Dec 7 '18 at 19:18










  • $begingroup$
    And the reason it can't be written as a finite union of convex sets would be, for example, that each point $(n+frac{1}{2}, 1)$ would have to be in a different one of the convex sets?
    $endgroup$
    – Daniel Schepler
    Dec 7 '18 at 22:04










  • $begingroup$
    @DanielSchepler Yes, that's it.
    $endgroup$
    – José Carlos Santos
    Dec 7 '18 at 22:23


















  • $begingroup$
    Why is this connected? Its a $xy$-plane separated by vertical lines $x = n$.
    $endgroup$
    – GNUSupporter 8964民主女神 地下教會
    Dec 7 '18 at 19:14










  • $begingroup$
    They're not lines. They're half-lines. It's the plane minus the vertical half-lines starting on points of the type $(n,0)$ and going up from there.
    $endgroup$
    – José Carlos Santos
    Dec 7 '18 at 19:18










  • $begingroup$
    And the reason it can't be written as a finite union of convex sets would be, for example, that each point $(n+frac{1}{2}, 1)$ would have to be in a different one of the convex sets?
    $endgroup$
    – Daniel Schepler
    Dec 7 '18 at 22:04










  • $begingroup$
    @DanielSchepler Yes, that's it.
    $endgroup$
    – José Carlos Santos
    Dec 7 '18 at 22:23
















$begingroup$
Why is this connected? Its a $xy$-plane separated by vertical lines $x = n$.
$endgroup$
– GNUSupporter 8964民主女神 地下教會
Dec 7 '18 at 19:14




$begingroup$
Why is this connected? Its a $xy$-plane separated by vertical lines $x = n$.
$endgroup$
– GNUSupporter 8964民主女神 地下教會
Dec 7 '18 at 19:14












$begingroup$
They're not lines. They're half-lines. It's the plane minus the vertical half-lines starting on points of the type $(n,0)$ and going up from there.
$endgroup$
– José Carlos Santos
Dec 7 '18 at 19:18




$begingroup$
They're not lines. They're half-lines. It's the plane minus the vertical half-lines starting on points of the type $(n,0)$ and going up from there.
$endgroup$
– José Carlos Santos
Dec 7 '18 at 19:18












$begingroup$
And the reason it can't be written as a finite union of convex sets would be, for example, that each point $(n+frac{1}{2}, 1)$ would have to be in a different one of the convex sets?
$endgroup$
– Daniel Schepler
Dec 7 '18 at 22:04




$begingroup$
And the reason it can't be written as a finite union of convex sets would be, for example, that each point $(n+frac{1}{2}, 1)$ would have to be in a different one of the convex sets?
$endgroup$
– Daniel Schepler
Dec 7 '18 at 22:04












$begingroup$
@DanielSchepler Yes, that's it.
$endgroup$
– José Carlos Santos
Dec 7 '18 at 22:23




$begingroup$
@DanielSchepler Yes, that's it.
$endgroup$
– José Carlos Santos
Dec 7 '18 at 22:23


















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