A property of the n-simplex
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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
asked Dec 2 '18 at 4:30
Juan Daniel Valdivia FuentesJuan Daniel Valdivia Fuentes
104
$endgroup$
add a comment |
$begingroup$
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
asked Dec 2 '18 at 4:30
Juan Daniel Valdivia FuentesJuan Daniel Valdivia Fuentes
104
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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
asked Dec 2 '18 at 4:30
Juan Daniel Valdivia FuentesJuan Daniel Valdivia Fuentes
104
$endgroup$
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
asked Dec 2 '18 at 4:30
Juan Daniel Valdivia FuentesJuan Daniel Valdivia Fuentes
104
asked Dec 2 '18 at 4:30
Juan Daniel Valdivia FuentesJuan Daniel Valdivia Fuentes
104
asked Dec 2 '18 at 4:30
Juan Daniel Valdivia FuentesJuan Daniel Valdivia Fuentes
104
asked Dec 2 '18 at 4:30
Juan Daniel Valdivia FuentesJuan Daniel Valdivia Fuentes
104
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
add a comment |
$begingroup$
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
add a comment |
1 Answer
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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 $Delta^n$.
answered Dec 2 '18 at 6:25
Henno BrandsmaHenno Brandsma
106k347114
$endgroup$
$begingroup$
Thanks !! i missed put in $Delta^{n}$ .
$endgroup$
– Juan Daniel Valdivia Fuentes
Dec 2 '18 at 13:56
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@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
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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 $Delta^n$.
answered Dec 2 '18 at 6:25
Henno BrandsmaHenno Brandsma
106k347114
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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 $Delta^n$.
answered Dec 2 '18 at 6:25
Henno BrandsmaHenno Brandsma
106k347114
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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 $Delta^n$.
answered Dec 2 '18 at 6:25
Henno BrandsmaHenno Brandsma
106k347114
$endgroup$
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 $Delta^n$.
answered Dec 2 '18 at 6:25
Henno BrandsmaHenno Brandsma
106k347114
answered Dec 2 '18 at 6:25
Henno BrandsmaHenno Brandsma
106k347114
answered Dec 2 '18 at 6:25
Henno BrandsmaHenno Brandsma
106k347114
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
add a comment |
$begingroup$
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
add a comment |
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$begingroup$
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