Mixing
Limit point and interior point
$begingroup$
Is any interior point also a limit point?
Judging from the definition, I believe every interior point is a limit point, but I'm not sure about it. If this is wrong, could you give me a counterexample?
(Since an interior point $p$ of a set $E$ has a neighborhood $N$ with radius $r$ such that $N$ is a subset of $E$. Obviously any neighborhood of $p$ with radius less than $r$ is a subset of $E$. Also, any neighborhood with radius greater than $r$ contains $N$ as a subset, so (I think) it is a limit point.)
general-topology
edited Aug 2 '12 at 4:31
enzotib
5,85321531
asked Aug 1 '12 at 21:59
TenguTengu
5331617
$endgroup$
add a comment |
$begingroup$
Is any interior point also a limit point?
Judging from the definition, I believe every interior point is a limit point, but I'm not sure about it. If this is wrong, could you give me a counterexample?
(Since an interior point $p$ of a set $E$ has a neighborhood $N$ with radius $r$ such that $N$ is a subset of $E$. Obviously any neighborhood of $p$ with radius less than $r$ is a subset of $E$. Also, any neighborhood with radius greater than $r$ contains $N$ as a subset, so (I think) it is a limit point.)
general-topology
edited Aug 2 '12 at 4:31
enzotib
5,85321531
asked Aug 1 '12 at 21:59
TenguTengu
5331617
$endgroup$
$begingroup$
By the way, you tagged this "general-topology", but speak of radii of nhoods. Are you working in metric spaces?
$endgroup$
– David Mitra
Aug 1 '12 at 22:06
$begingroup$
Yes. I just started analysis (Walter Rudin The principle of mathematical analysis) and I came up with this question when reading the chapter named the basic topology
$endgroup$
– Tengu
Aug 1 '12 at 22:09
add a comment |
$begingroup$
Is any interior point also a limit point?
Judging from the definition, I believe every interior point is a limit point, but I'm not sure about it. If this is wrong, could you give me a counterexample?
(Since an interior point $p$ of a set $E$ has a neighborhood $N$ with radius $r$ such that $N$ is a subset of $E$. Obviously any neighborhood of $p$ with radius less than $r$ is a subset of $E$. Also, any neighborhood with radius greater than $r$ contains $N$ as a subset, so (I think) it is a limit point.)
general-topology
edited Aug 2 '12 at 4:31
enzotib
5,85321531
asked Aug 1 '12 at 21:59
TenguTengu
5331617
$endgroup$
Is any interior point also a limit point?
Judging from the definition, I believe every interior point is a limit point, but I'm not sure about it. If this is wrong, could you give me a counterexample?
(Since an interior point $p$ of a set $E$ has a neighborhood $N$ with radius $r$ such that $N$ is a subset of $E$. Obviously any neighborhood of $p$ with radius less than $r$ is a subset of $E$. Also, any neighborhood with radius greater than $r$ contains $N$ as a subset, so (I think) it is a limit point.)
general-topology
general-topology
edited Aug 2 '12 at 4:31
enzotib
5,85321531
asked Aug 1 '12 at 21:59
TenguTengu
5331617
edited Aug 2 '12 at 4:31
enzotib
5,85321531
asked Aug 1 '12 at 21:59
TenguTengu
5331617
edited Aug 2 '12 at 4:31
enzotib
5,85321531
edited Aug 2 '12 at 4:31
enzotib
5,85321531
edited Aug 2 '12 at 4:31
enzotib
5,85321531
5,85321531
asked Aug 1 '12 at 21:59
TenguTengu
5331617
asked Aug 1 '12 at 21:59
TenguTengu
5331617
asked Aug 1 '12 at 21:59
TenguTengu
5331617
5331617
$begingroup$
By the way, you tagged this "general-topology", but speak of radii of nhoods. Are you working in metric spaces?
$endgroup$
– David Mitra
Aug 1 '12 at 22:06
$begingroup$
Yes. I just started analysis (Walter Rudin The principle of mathematical analysis) and I came up with this question when reading the chapter named the basic topology
$endgroup$
– Tengu
Aug 1 '12 at 22:09
add a comment |
$begingroup$
By the way, you tagged this "general-topology", but speak of radii of nhoods. Are you working in metric spaces?
$endgroup$
– David Mitra
Aug 1 '12 at 22:06
$begingroup$
Yes. I just started analysis (Walter Rudin The principle of mathematical analysis) and I came up with this question when reading the chapter named the basic topology
$endgroup$
– Tengu
Aug 1 '12 at 22:09
$begingroup$
By the way, you tagged this "general-topology", but speak of radii of nhoods. Are you working in metric spaces?
$endgroup$
– David Mitra
Aug 1 '12 at 22:06
$begingroup$
By the way, you tagged this "general-topology", but speak of radii of nhoods. Are you working in metric spaces?
$endgroup$
– David Mitra
Aug 1 '12 at 22:06
$begingroup$
Yes. I just started analysis (Walter Rudin The principle of mathematical analysis) and I came up with this question when reading the chapter named the basic topology
$endgroup$
– Tengu
Aug 1 '12 at 22:09
$begingroup$
Yes. I just started analysis (Walter Rudin The principle of mathematical analysis) and I came up with this question when reading the chapter named the basic topology
$endgroup$
– Tengu
Aug 1 '12 at 22:09
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
A discrete space has no limit points, but every point is an interior point.
answered Aug 1 '12 at 22:17
InkInk
4,46611017
$endgroup$
$begingroup$
Brilliant answer, Ink!
$endgroup$
– WishingFish
Jul 28 '13 at 3:13
add a comment |
$begingroup$
No an interior point is not a limit point in general. Suppose you have $mathbb{N}$ with the discrete metric. The point $1$ is a interior point since ${1}$ is an open set. However, $1$ is not a limit point of $mathbb{N}$ because ${1}$ is an open neighborhood of $1$ which does not intersect $mathbb{N}$ in any point beside $1$.
answered Aug 1 '12 at 22:03
WilliamWilliam
17.3k22256
$endgroup$
$begingroup$
Is a discrete space a space all of whose points are isolated points? (I could not find it in my textbook) Anyway I don't quite understand why 1 is a interior point of N...
$endgroup$
– Tengu
Aug 1 '12 at 22:16
$begingroup$
A discrete topology is a space where every subset is open. Topologically equivalent, it is a space endowed with the discrete metric. Since every subset is open, the set ${1}$ is open.
$endgroup$
– William
Aug 1 '12 at 22:18
$begingroup$
Thank you very much. I think I should study more in topology.
$endgroup$
– Tengu
Aug 1 '12 at 22:23
$begingroup$
@Tengu in the language of Rudin, the set of points in $mathbb{N}$ whose distance from 1 is less than, say, $frac{1}{2}$, is exactly ${1}$, thus this single-element set is an open ball (and thus open).
$endgroup$
– MartianInvader
Aug 1 '12 at 22:23
$begingroup$
Thank you for your reply. Since the neighborhood of 1 with radius 1/2 does not include any point except 1 on N, 1 is an open subset of N?
$endgroup$
– Tengu
Aug 2 '12 at 23:09
add a comment |
Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
A discrete space has no limit points, but every point is an interior point.
answered Aug 1 '12 at 22:17
InkInk
4,46611017
$endgroup$
$begingroup$
Brilliant answer, Ink!
$endgroup$
– WishingFish
Jul 28 '13 at 3:13
add a comment |
$begingroup$
A discrete space has no limit points, but every point is an interior point.
answered Aug 1 '12 at 22:17
InkInk
4,46611017
$endgroup$
$begingroup$
Brilliant answer, Ink!
$endgroup$
– WishingFish
Jul 28 '13 at 3:13
add a comment |
$begingroup$
A discrete space has no limit points, but every point is an interior point.
answered Aug 1 '12 at 22:17
InkInk
4,46611017
$endgroup$
A discrete space has no limit points, but every point is an interior point.
answered Aug 1 '12 at 22:17
InkInk
4,46611017
answered Aug 1 '12 at 22:17
InkInk
4,46611017
answered Aug 1 '12 at 22:17
InkInk
4,46611017
answered Aug 1 '12 at 22:17
InkInk
4,46611017
4,46611017
$begingroup$
Brilliant answer, Ink!
$endgroup$
– WishingFish
Jul 28 '13 at 3:13
add a comment |
$begingroup$
Brilliant answer, Ink!
$endgroup$
– WishingFish
Jul 28 '13 at 3:13
$begingroup$
Brilliant answer, Ink!
$endgroup$
– WishingFish
Jul 28 '13 at 3:13
$begingroup$
Brilliant answer, Ink!
$endgroup$
– WishingFish
Jul 28 '13 at 3:13
add a comment |
$begingroup$
No an interior point is not a limit point in general. Suppose you have $mathbb{N}$ with the discrete metric. The point $1$ is a interior point since ${1}$ is an open set. However, $1$ is not a limit point of $mathbb{N}$ because ${1}$ is an open neighborhood of $1$ which does not intersect $mathbb{N}$ in any point beside $1$.
answered Aug 1 '12 at 22:03
WilliamWilliam
17.3k22256
$endgroup$
$begingroup$
Is a discrete space a space all of whose points are isolated points? (I could not find it in my textbook) Anyway I don't quite understand why 1 is a interior point of N...
$endgroup$
– Tengu
Aug 1 '12 at 22:16
$begingroup$
A discrete topology is a space where every subset is open. Topologically equivalent, it is a space endowed with the discrete metric. Since every subset is open, the set ${1}$ is open.
$endgroup$
– William
Aug 1 '12 at 22:18
$begingroup$
Thank you very much. I think I should study more in topology.
$endgroup$
– Tengu
Aug 1 '12 at 22:23
$begingroup$
@Tengu in the language of Rudin, the set of points in $mathbb{N}$ whose distance from 1 is less than, say, $frac{1}{2}$, is exactly ${1}$, thus this single-element set is an open ball (and thus open).
$endgroup$
– MartianInvader
Aug 1 '12 at 22:23
$begingroup$
Thank you for your reply. Since the neighborhood of 1 with radius 1/2 does not include any point except 1 on N, 1 is an open subset of N?
$endgroup$
– Tengu
Aug 2 '12 at 23:09
add a comment |
$begingroup$
No an interior point is not a limit point in general. Suppose you have $mathbb{N}$ with the discrete metric. The point $1$ is a interior point since ${1}$ is an open set. However, $1$ is not a limit point of $mathbb{N}$ because ${1}$ is an open neighborhood of $1$ which does not intersect $mathbb{N}$ in any point beside $1$.
answered Aug 1 '12 at 22:03
WilliamWilliam
17.3k22256
$endgroup$
$begingroup$
Is a discrete space a space all of whose points are isolated points? (I could not find it in my textbook) Anyway I don't quite understand why 1 is a interior point of N...
$endgroup$
– Tengu
Aug 1 '12 at 22:16
$begingroup$
A discrete topology is a space where every subset is open. Topologically equivalent, it is a space endowed with the discrete metric. Since every subset is open, the set ${1}$ is open.
$endgroup$
– William
Aug 1 '12 at 22:18
$begingroup$
Thank you very much. I think I should study more in topology.
$endgroup$
– Tengu
Aug 1 '12 at 22:23
$begingroup$
@Tengu in the language of Rudin, the set of points in $mathbb{N}$ whose distance from 1 is less than, say, $frac{1}{2}$, is exactly ${1}$, thus this single-element set is an open ball (and thus open).
$endgroup$
– MartianInvader
Aug 1 '12 at 22:23
$begingroup$
Thank you for your reply. Since the neighborhood of 1 with radius 1/2 does not include any point except 1 on N, 1 is an open subset of N?
$endgroup$
– Tengu
Aug 2 '12 at 23:09
add a comment |
$begingroup$
No an interior point is not a limit point in general. Suppose you have $mathbb{N}$ with the discrete metric. The point $1$ is a interior point since ${1}$ is an open set. However, $1$ is not a limit point of $mathbb{N}$ because ${1}$ is an open neighborhood of $1$ which does not intersect $mathbb{N}$ in any point beside $1$.
answered Aug 1 '12 at 22:03
WilliamWilliam
17.3k22256
$endgroup$
No an interior point is not a limit point in general. Suppose you have $mathbb{N}$ with the discrete metric. The point $1$ is a interior point since ${1}$ is an open set. However, $1$ is not a limit point of $mathbb{N}$ because ${1}$ is an open neighborhood of $1$ which does not intersect $mathbb{N}$ in any point beside $1$.
answered Aug 1 '12 at 22:03
WilliamWilliam
17.3k22256
answered Aug 1 '12 at 22:03
WilliamWilliam
17.3k22256
answered Aug 1 '12 at 22:03
WilliamWilliam
17.3k22256
answered Aug 1 '12 at 22:03
WilliamWilliam
17.3k22256
17.3k22256
$begingroup$
Is a discrete space a space all of whose points are isolated points? (I could not find it in my textbook) Anyway I don't quite understand why 1 is a interior point of N...
$endgroup$
– Tengu
Aug 1 '12 at 22:16
$begingroup$
A discrete topology is a space where every subset is open. Topologically equivalent, it is a space endowed with the discrete metric. Since every subset is open, the set ${1}$ is open.
$endgroup$
– William
Aug 1 '12 at 22:18
$begingroup$
Thank you very much. I think I should study more in topology.
$endgroup$
– Tengu
Aug 1 '12 at 22:23
$begingroup$
@Tengu in the language of Rudin, the set of points in $mathbb{N}$ whose distance from 1 is less than, say, $frac{1}{2}$, is exactly ${1}$, thus this single-element set is an open ball (and thus open).
$endgroup$
– MartianInvader
Aug 1 '12 at 22:23
$begingroup$
Thank you for your reply. Since the neighborhood of 1 with radius 1/2 does not include any point except 1 on N, 1 is an open subset of N?
$endgroup$
– Tengu
Aug 2 '12 at 23:09
add a comment |
$begingroup$
Is a discrete space a space all of whose points are isolated points? (I could not find it in my textbook) Anyway I don't quite understand why 1 is a interior point of N...
$endgroup$
– Tengu
Aug 1 '12 at 22:16
$begingroup$
A discrete topology is a space where every subset is open. Topologically equivalent, it is a space endowed with the discrete metric. Since every subset is open, the set ${1}$ is open.
$endgroup$
– William
Aug 1 '12 at 22:18
$begingroup$
Thank you very much. I think I should study more in topology.
$endgroup$
– Tengu
Aug 1 '12 at 22:23
$begingroup$
@Tengu in the language of Rudin, the set of points in $mathbb{N}$ whose distance from 1 is less than, say, $frac{1}{2}$, is exactly ${1}$, thus this single-element set is an open ball (and thus open).
$endgroup$
– MartianInvader
Aug 1 '12 at 22:23
$begingroup$
Thank you for your reply. Since the neighborhood of 1 with radius 1/2 does not include any point except 1 on N, 1 is an open subset of N?
$endgroup$
– Tengu
Aug 2 '12 at 23:09
$begingroup$
Is a discrete space a space all of whose points are isolated points? (I could not find it in my textbook) Anyway I don't quite understand why 1 is a interior point of N...
$endgroup$
– Tengu
Aug 1 '12 at 22:16
$begingroup$
Is a discrete space a space all of whose points are isolated points? (I could not find it in my textbook) Anyway I don't quite understand why 1 is a interior point of N...
$endgroup$
– Tengu
Aug 1 '12 at 22:16
$begingroup$
A discrete topology is a space where every subset is open. Topologically equivalent, it is a space endowed with the discrete metric. Since every subset is open, the set ${1}$ is open.
$endgroup$
– William
Aug 1 '12 at 22:18
$begingroup$
A discrete topology is a space where every subset is open. Topologically equivalent, it is a space endowed with the discrete metric. Since every subset is open, the set ${1}$ is open.
$endgroup$
– William
Aug 1 '12 at 22:18
$begingroup$
Thank you very much. I think I should study more in topology.
$endgroup$
– Tengu
Aug 1 '12 at 22:23
$begingroup$
Thank you very much. I think I should study more in topology.
$endgroup$
– Tengu
Aug 1 '12 at 22:23
$begingroup$
@Tengu in the language of Rudin, the set of points in $mathbb{N}$ whose distance from 1 is less than, say, $frac{1}{2}$, is exactly ${1}$, thus this single-element set is an open ball (and thus open).
$endgroup$
– MartianInvader
Aug 1 '12 at 22:23
$begingroup$
@Tengu in the language of Rudin, the set of points in $mathbb{N}$ whose distance from 1 is less than, say, $frac{1}{2}$, is exactly ${1}$, thus this single-element set is an open ball (and thus open).
$endgroup$
– MartianInvader
Aug 1 '12 at 22:23
$begingroup$
Thank you for your reply. Since the neighborhood of 1 with radius 1/2 does not include any point except 1 on N, 1 is an open subset of N?
$endgroup$
– Tengu
Aug 2 '12 at 23:09
$begingroup$
Thank you for your reply. Since the neighborhood of 1 with radius 1/2 does not include any point except 1 on N, 1 is an open subset of N?
$endgroup$
– Tengu
Aug 2 '12 at 23:09
add a comment |
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$begingroup$
By the way, you tagged this "general-topology", but speak of radii of nhoods. Are you working in metric spaces?
$endgroup$
– David Mitra
Aug 1 '12 at 22:06
$begingroup$
Yes. I just started analysis (Walter Rudin The principle of mathematical analysis) and I came up with this question when reading the chapter named the basic topology
$endgroup$
– Tengu
Aug 1 '12 at 22:09