A property of the n-simplex












1












$begingroup$


Let $Delta ^{n}$ a n-simplex and $Delta_{0}^{n-1}, Delta_{1}^{n-1}, cdots , Delta_{n}^{n-1}$ be the $(n-1)$ - faces of $Delta^{n}$.



The subsets $Delta^{n} backslash Delta_{i}^{n-1}$ are open for all $i$ ?






general-topology convex-analysis







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$endgroup$












  • $begingroup$
    No. If you remove an edge from a two-simplex (triangle), the resulting figure is neither open nor closed.
    $endgroup$
    – saulspatz
    Dec 2 '18 at 4:59
















1












$begingroup$


Let $Delta ^{n}$ a n-simplex and $Delta_{0}^{n-1}, Delta_{1}^{n-1}, cdots , Delta_{n}^{n-1}$ be the $(n-1)$ - faces of $Delta^{n}$.



The subsets $Delta^{n} backslash Delta_{i}^{n-1}$ are open for all $i$ ?






general-topology convex-analysis







share|cite|improve this question









$endgroup$












  • $begingroup$
    No. If you remove an edge from a two-simplex (triangle), the resulting figure is neither open nor closed.
    $endgroup$
    – saulspatz
    Dec 2 '18 at 4:59














1












1








1





$begingroup$


Let $Delta ^{n}$ a n-simplex and $Delta_{0}^{n-1}, Delta_{1}^{n-1}, cdots , Delta_{n}^{n-1}$ be the $(n-1)$ - faces of $Delta^{n}$.



The subsets $Delta^{n} backslash Delta_{i}^{n-1}$ are open for all $i$ ?






general-topology convex-analysis







share|cite|improve this question









$endgroup$




Let $Delta ^{n}$ a n-simplex and $Delta_{0}^{n-1}, Delta_{1}^{n-1}, cdots , Delta_{n}^{n-1}$ be the $(n-1)$ - faces of $Delta^{n}$.



The subsets $Delta^{n} backslash Delta_{i}^{n-1}$ are open for all $i$ ?






general-topology convex-analysis




general-topology convex-analysis






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 2 '18 at 4:30









Juan Daniel Valdivia FuentesJuan Daniel Valdivia Fuentes

104




104












  • $begingroup$
    No. If you remove an edge from a two-simplex (triangle), the resulting figure is neither open nor closed.
    $endgroup$
    – saulspatz
    Dec 2 '18 at 4:59


















  • $begingroup$
    No. If you remove an edge from a two-simplex (triangle), the resulting figure is neither open nor closed.
    $endgroup$
    – saulspatz
    Dec 2 '18 at 4:59
















$begingroup$
No. If you remove an edge from a two-simplex (triangle), the resulting figure is neither open nor closed.
$endgroup$
– saulspatz
Dec 2 '18 at 4:59




$begingroup$
No. If you remove an edge from a two-simplex (triangle), the resulting figure is neither open nor closed.
$endgroup$
– saulspatz
Dec 2 '18 at 4:59










1 Answer
1






active

oldest

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0












$begingroup$

Yes, these sets are open (in $Delta^n$!) as all (sub)simplices are compact and thus absolutely closed. And $Delta^n setminus Delta_i^{n-1} = Delta^n cap (mathbb{R}^{n+1}setminus Delta_{i}^{n-1})$, which is the intersection of an open set in the ambient space with $Delta^n$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks !! i missed put in $Delta^{n}$ .
    $endgroup$
    – Juan Daniel Valdivia Fuentes
    Dec 2 '18 at 13:56










  • $begingroup$
    @JuanDanielValdiviaFuentes They are neither open nor closed in the ambient space, but inside the lareg simplex they are open.
    $endgroup$
    – Henno Brandsma
    Dec 2 '18 at 14:00











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1 Answer
1






active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









0












$begingroup$

Yes, these sets are open (in $Delta^n$!) as all (sub)simplices are compact and thus absolutely closed. And $Delta^n setminus Delta_i^{n-1} = Delta^n cap (mathbb{R}^{n+1}setminus Delta_{i}^{n-1})$, which is the intersection of an open set in the ambient space with $Delta^n$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks !! i missed put in $Delta^{n}$ .
    $endgroup$
    – Juan Daniel Valdivia Fuentes
    Dec 2 '18 at 13:56










  • $begingroup$
    @JuanDanielValdiviaFuentes They are neither open nor closed in the ambient space, but inside the lareg simplex they are open.
    $endgroup$
    – Henno Brandsma
    Dec 2 '18 at 14:00
















0












$begingroup$

Yes, these sets are open (in $Delta^n$!) as all (sub)simplices are compact and thus absolutely closed. And $Delta^n setminus Delta_i^{n-1} = Delta^n cap (mathbb{R}^{n+1}setminus Delta_{i}^{n-1})$, which is the intersection of an open set in the ambient space with $Delta^n$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks !! i missed put in $Delta^{n}$ .
    $endgroup$
    – Juan Daniel Valdivia Fuentes
    Dec 2 '18 at 13:56










  • $begingroup$
    @JuanDanielValdiviaFuentes They are neither open nor closed in the ambient space, but inside the lareg simplex they are open.
    $endgroup$
    – Henno Brandsma
    Dec 2 '18 at 14:00














0












0








0





$begingroup$

Yes, these sets are open (in $Delta^n$!) as all (sub)simplices are compact and thus absolutely closed. And $Delta^n setminus Delta_i^{n-1} = Delta^n cap (mathbb{R}^{n+1}setminus Delta_{i}^{n-1})$, which is the intersection of an open set in the ambient space with $Delta^n$.






share|cite|improve this answer









$endgroup$



Yes, these sets are open (in $Delta^n$!) as all (sub)simplices are compact and thus absolutely closed. And $Delta^n setminus Delta_i^{n-1} = Delta^n cap (mathbb{R}^{n+1}setminus Delta_{i}^{n-1})$, which is the intersection of an open set in the ambient space with $Delta^n$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 2 '18 at 6:25









Henno BrandsmaHenno Brandsma

106k347114




106k347114












  • $begingroup$
    Thanks !! i missed put in $Delta^{n}$ .
    $endgroup$
    – Juan Daniel Valdivia Fuentes
    Dec 2 '18 at 13:56










  • $begingroup$
    @JuanDanielValdiviaFuentes They are neither open nor closed in the ambient space, but inside the lareg simplex they are open.
    $endgroup$
    – Henno Brandsma
    Dec 2 '18 at 14:00


















  • $begingroup$
    Thanks !! i missed put in $Delta^{n}$ .
    $endgroup$
    – Juan Daniel Valdivia Fuentes
    Dec 2 '18 at 13:56










  • $begingroup$
    @JuanDanielValdiviaFuentes They are neither open nor closed in the ambient space, but inside the lareg simplex they are open.
    $endgroup$
    – Henno Brandsma
    Dec 2 '18 at 14:00
















$begingroup$
Thanks !! i missed put in $Delta^{n}$ .
$endgroup$
– Juan Daniel Valdivia Fuentes
Dec 2 '18 at 13:56




$begingroup$
Thanks !! i missed put in $Delta^{n}$ .
$endgroup$
– Juan Daniel Valdivia Fuentes
Dec 2 '18 at 13:56












$begingroup$
@JuanDanielValdiviaFuentes They are neither open nor closed in the ambient space, but inside the lareg simplex they are open.
$endgroup$
– Henno Brandsma
Dec 2 '18 at 14:00




$begingroup$
@JuanDanielValdiviaFuentes They are neither open nor closed in the ambient space, but inside the lareg simplex they are open.
$endgroup$
– Henno Brandsma
Dec 2 '18 at 14:00


















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