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Checking if a set is open and relatively open in the sphere $x^2 + y^2 + z^2 = 1$
Is a sphere a closed set?Is this set relatively open or closed?Taxicab metric with open, closed unit ball, and unit sphere.The Geodesics of a Sphere.Parametrization of the sphere and the torus.Is $(0,1) bigcap mathbbQ$ an open set in the standard topology of $mathbbR$?Prove torus is homeomorphic to $mathbb S^1times mathbb S^1$Checking if a set is closed / openParameterizing an $m$-covered sphereTrouble with Spherical Coordinates and the Transformation Formula
$begingroup$
Let S be the sphere $(x,y,z) in mathbb R^3 mid x^2 + y^2 + z^2 = 1$.
First I want to check if the set $A = (cosphi sintheta, sinphi sintheta, costheta ) midphi in (0,pi), theta in (0,pi)$ is open in $mathbb R^3$.
I am pretty sure that it isn't open since it is half of the sphere S, and for the example for the point $(0,1,0) in A$ we can't find a ball in $A$ that surrounds it.
The second task is to find whether $A$ is relatively open in $S$.
Here I am not so sure but I think I can take the open set: $V = $ $B(0,2)$ with $y>0$ and then $S bigcap V = A$ which proves that it is relatively open in $S$.
However the set $B = (cosphi sintheta, sinphi sintheta, costheta ) midphi in [0,pi], theta in [0,pi]$ is not relatively open in $S$, right?
Am I correct here?
Help would be appreciated.
real-analysis calculus general-topology
$endgroup$
add a comment |
$begingroup$
Let S be the sphere $(x,y,z) in mathbb R^3 mid x^2 + y^2 + z^2 = 1$.
First I want to check if the set $A = (cosphi sintheta, sinphi sintheta, costheta ) midphi in (0,pi), theta in (0,pi)$ is open in $mathbb R^3$.
I am pretty sure that it isn't open since it is half of the sphere S, and for the example for the point $(0,1,0) in A$ we can't find a ball in $A$ that surrounds it.
The second task is to find whether $A$ is relatively open in $S$.
Here I am not so sure but I think I can take the open set: $V = $ $B(0,2)$ with $y>0$ and then $S bigcap V = A$ which proves that it is relatively open in $S$.
However the set $B = (cosphi sintheta, sinphi sintheta, costheta ) midphi in [0,pi], theta in [0,pi]$ is not relatively open in $S$, right?
Am I correct here?
Help would be appreciated.
real-analysis calculus general-topology
$endgroup$
1
$begingroup$
It seems OK.$$
$endgroup$
– Giuseppe Negro
2 days ago
$begingroup$
Thanks. But how can I formally prove that $B$ is not relatively open?
$endgroup$
– Gabi G
2 days ago
$begingroup$
You already did, I think it is enough.
$endgroup$
– Giuseppe Negro
2 days ago
$begingroup$
I proved that A is. In the comment I am talking about B
$endgroup$
– Gabi G
2 days ago
add a comment |
$begingroup$
Let S be the sphere $(x,y,z) in mathbb R^3 mid x^2 + y^2 + z^2 = 1$.
First I want to check if the set $A = (cosphi sintheta, sinphi sintheta, costheta ) midphi in (0,pi), theta in (0,pi)$ is open in $mathbb R^3$.
I am pretty sure that it isn't open since it is half of the sphere S, and for the example for the point $(0,1,0) in A$ we can't find a ball in $A$ that surrounds it.
The second task is to find whether $A$ is relatively open in $S$.
Here I am not so sure but I think I can take the open set: $V = $ $B(0,2)$ with $y>0$ and then $S bigcap V = A$ which proves that it is relatively open in $S$.
However the set $B = (cosphi sintheta, sinphi sintheta, costheta ) midphi in [0,pi], theta in [0,pi]$ is not relatively open in $S$, right?
Am I correct here?
Help would be appreciated.
real-analysis calculus general-topology
$endgroup$
Let S be the sphere $(x,y,z) in mathbb R^3 mid x^2 + y^2 + z^2 = 1$.
First I want to check if the set $A = (cosphi sintheta, sinphi sintheta, costheta ) midphi in (0,pi), theta in (0,pi)$ is open in $mathbb R^3$.
I am pretty sure that it isn't open since it is half of the sphere S, and for the example for the point $(0,1,0) in A$ we can't find a ball in $A$ that surrounds it.
The second task is to find whether $A$ is relatively open in $S$.
Here I am not so sure but I think I can take the open set: $V = $ $B(0,2)$ with $y>0$ and then $S bigcap V = A$ which proves that it is relatively open in $S$.
However the set $B = (cosphi sintheta, sinphi sintheta, costheta ) midphi in [0,pi], theta in [0,pi]$ is not relatively open in $S$, right?
Am I correct here?
Help would be appreciated.
real-analysis calculus general-topology
real-analysis calculus general-topology
edited 2 days ago
Jean Marie
30.6k42154
30.6k42154
asked 2 days ago
Gabi GGabi G
442110
442110
1
$begingroup$
It seems OK.$$
$endgroup$
– Giuseppe Negro
2 days ago
$begingroup$
Thanks. But how can I formally prove that $B$ is not relatively open?
$endgroup$
– Gabi G
2 days ago
$begingroup$
You already did, I think it is enough.
$endgroup$
– Giuseppe Negro
2 days ago
$begingroup$
I proved that A is. In the comment I am talking about B
$endgroup$
– Gabi G
2 days ago
add a comment |
1
$begingroup$
It seems OK.$$
$endgroup$
– Giuseppe Negro
2 days ago
$begingroup$
Thanks. But how can I formally prove that $B$ is not relatively open?
$endgroup$
– Gabi G
2 days ago
$begingroup$
You already did, I think it is enough.
$endgroup$
– Giuseppe Negro
2 days ago
$begingroup$
I proved that A is. In the comment I am talking about B
$endgroup$
– Gabi G
2 days ago
1
1
$begingroup$
It seems OK.$$
$endgroup$
– Giuseppe Negro
2 days ago
$begingroup$
It seems OK.$$
$endgroup$
– Giuseppe Negro
2 days ago
$begingroup$
Thanks. But how can I formally prove that $B$ is not relatively open?
$endgroup$
– Gabi G
2 days ago
$begingroup$
Thanks. But how can I formally prove that $B$ is not relatively open?
$endgroup$
– Gabi G
2 days ago
$begingroup$
You already did, I think it is enough.
$endgroup$
– Giuseppe Negro
2 days ago
$begingroup$
You already did, I think it is enough.
$endgroup$
– Giuseppe Negro
2 days ago
$begingroup$
I proved that A is. In the comment I am talking about B
$endgroup$
– Gabi G
2 days ago
$begingroup$
I proved that A is. In the comment I am talking about B
$endgroup$
– Gabi G
2 days ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Suppose $U$ is an open set in $mathbbR^3$ so that $Ucap S=B$. Then since $U$ is open, there is a ball $D$ around, say, the point $(0,0,1)$ with radius $epsilon$ that is contained inside $U$. But since $Ucap S=B$ and $Dsubset U$, it follows that $Dcap Ssubset B$. In other words, the small ball $D$ centered at the north pole intersects the sphere in some open set $Dcap S$ which will be contained in the closed half-sphere $B$. But this is impossible, you can find points in the intersection $Dcap S$ which have $theta's$ that fall outside of $0leqthetaleq pi$.
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add a comment |
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$begingroup$
Suppose $U$ is an open set in $mathbbR^3$ so that $Ucap S=B$. Then since $U$ is open, there is a ball $D$ around, say, the point $(0,0,1)$ with radius $epsilon$ that is contained inside $U$. But since $Ucap S=B$ and $Dsubset U$, it follows that $Dcap Ssubset B$. In other words, the small ball $D$ centered at the north pole intersects the sphere in some open set $Dcap S$ which will be contained in the closed half-sphere $B$. But this is impossible, you can find points in the intersection $Dcap S$ which have $theta's$ that fall outside of $0leqthetaleq pi$.
$endgroup$
add a comment |
$begingroup$
Suppose $U$ is an open set in $mathbbR^3$ so that $Ucap S=B$. Then since $U$ is open, there is a ball $D$ around, say, the point $(0,0,1)$ with radius $epsilon$ that is contained inside $U$. But since $Ucap S=B$ and $Dsubset U$, it follows that $Dcap Ssubset B$. In other words, the small ball $D$ centered at the north pole intersects the sphere in some open set $Dcap S$ which will be contained in the closed half-sphere $B$. But this is impossible, you can find points in the intersection $Dcap S$ which have $theta's$ that fall outside of $0leqthetaleq pi$.
$endgroup$
add a comment |
$begingroup$
Suppose $U$ is an open set in $mathbbR^3$ so that $Ucap S=B$. Then since $U$ is open, there is a ball $D$ around, say, the point $(0,0,1)$ with radius $epsilon$ that is contained inside $U$. But since $Ucap S=B$ and $Dsubset U$, it follows that $Dcap Ssubset B$. In other words, the small ball $D$ centered at the north pole intersects the sphere in some open set $Dcap S$ which will be contained in the closed half-sphere $B$. But this is impossible, you can find points in the intersection $Dcap S$ which have $theta's$ that fall outside of $0leqthetaleq pi$.
$endgroup$
Suppose $U$ is an open set in $mathbbR^3$ so that $Ucap S=B$. Then since $U$ is open, there is a ball $D$ around, say, the point $(0,0,1)$ with radius $epsilon$ that is contained inside $U$. But since $Ucap S=B$ and $Dsubset U$, it follows that $Dcap Ssubset B$. In other words, the small ball $D$ centered at the north pole intersects the sphere in some open set $Dcap S$ which will be contained in the closed half-sphere $B$. But this is impossible, you can find points in the intersection $Dcap S$ which have $theta's$ that fall outside of $0leqthetaleq pi$.
answered 2 days ago
PrototankPrototank
1,229920
1,229920
add a comment |
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1
$begingroup$
It seems OK.$$
$endgroup$
– Giuseppe Negro
2 days ago
$begingroup$
Thanks. But how can I formally prove that $B$ is not relatively open?
$endgroup$
– Gabi G
2 days ago
$begingroup$
You already did, I think it is enough.
$endgroup$
– Giuseppe Negro
2 days ago
$begingroup$
I proved that A is. In the comment I am talking about B
$endgroup$
– Gabi G
2 days ago