Is $0$ a cluster point of the set ${0}$?
$begingroup$
I just read the theorem
A subset of R is closed if and only if it contains all of its cluster points.
And I found that the set {0} is closed (Is that correct?), then the set {0} should contains all its cluster points, right? But I just checked the definition of cluster point in the textbook (it appears at the beginning of the limit of function) and found that 0 is not the cluster point of the set {0}
I want to know where the false is ?
real-analysis real-numbers
edited Dec 15 '18 at 7:13
anomaly
17.6k42666
asked Dec 15 '18 at 6:48
Yang GaoYang Gao
161
$endgroup$
|
show 1 more comment
$begingroup$
I just read the theorem
A subset of R is closed if and only if it contains all of its cluster points.
And I found that the set {0} is closed (Is that correct?), then the set {0} should contains all its cluster points, right? But I just checked the definition of cluster point in the textbook (it appears at the beginning of the limit of function) and found that 0 is not the cluster point of the set {0}
I want to know where the false is ?
real-analysis real-numbers
edited Dec 15 '18 at 7:13
anomaly
17.6k42666
asked Dec 15 '18 at 6:48
Yang GaoYang Gao
161
$endgroup$
$begingroup$
Do you think that the set is not closed? If yes then can you give a cluster point which is not in the set? The definition says that A set is closed iff it contains all its cluster point. If 0 is not the cluster point then why do you even worry about it?
$endgroup$
– Rakesh Bhatt
Dec 15 '18 at 6:51
$begingroup$
I got it ! I just thought 0 "should" be a cluster point of the set .Then the set does not have a cluster point, right? Thank you. But wait a minute, the set {0} contains the element 0 !
$endgroup$
– Yang Gao
Dec 15 '18 at 7:06
3
$begingroup$
Post the definition of "cluster point" in your book.
$endgroup$
– xbh
Dec 15 '18 at 7:12
$begingroup$
I think it appears at the answer provided by Anthony Ter .
$endgroup$
– Yang Gao
Dec 15 '18 at 7:19
$begingroup$
@YangGao The set {0} contains the element 0! Yes, It contains. So what?
$endgroup$
– Rakesh Bhatt
Dec 15 '18 at 9:21
|
show 1 more comment
$begingroup$
I just read the theorem
A subset of R is closed if and only if it contains all of its cluster points.
And I found that the set {0} is closed (Is that correct?), then the set {0} should contains all its cluster points, right? But I just checked the definition of cluster point in the textbook (it appears at the beginning of the limit of function) and found that 0 is not the cluster point of the set {0}
I want to know where the false is ?
real-analysis real-numbers
edited Dec 15 '18 at 7:13
anomaly
17.6k42666
asked Dec 15 '18 at 6:48
Yang GaoYang Gao
161
$endgroup$
I just read the theorem
A subset of R is closed if and only if it contains all of its cluster points.
And I found that the set {0} is closed (Is that correct?), then the set {0} should contains all its cluster points, right? But I just checked the definition of cluster point in the textbook (it appears at the beginning of the limit of function) and found that 0 is not the cluster point of the set {0}
I want to know where the false is ?
real-analysis real-numbers
real-analysis real-numbers
edited Dec 15 '18 at 7:13
anomaly
17.6k42666
asked Dec 15 '18 at 6:48
Yang GaoYang Gao
161
edited Dec 15 '18 at 7:13
anomaly
17.6k42666
asked Dec 15 '18 at 6:48
Yang GaoYang Gao
161
edited Dec 15 '18 at 7:13
anomaly
17.6k42666
edited Dec 15 '18 at 7:13
anomaly
17.6k42666
edited Dec 15 '18 at 7:13
anomaly
17.6k42666
17.6k42666
asked Dec 15 '18 at 6:48
Yang GaoYang Gao
161
asked Dec 15 '18 at 6:48
Yang GaoYang Gao
161
asked Dec 15 '18 at 6:48
Yang GaoYang Gao
161
161
$begingroup$
Do you think that the set is not closed? If yes then can you give a cluster point which is not in the set? The definition says that A set is closed iff it contains all its cluster point. If 0 is not the cluster point then why do you even worry about it?
$endgroup$
– Rakesh Bhatt
Dec 15 '18 at 6:51
$begingroup$
I got it ! I just thought 0 "should" be a cluster point of the set .Then the set does not have a cluster point, right? Thank you. But wait a minute, the set {0} contains the element 0 !
$endgroup$
– Yang Gao
Dec 15 '18 at 7:06
3
$begingroup$
Post the definition of "cluster point" in your book.
$endgroup$
– xbh
Dec 15 '18 at 7:12
$begingroup$
I think it appears at the answer provided by Anthony Ter .
$endgroup$
– Yang Gao
Dec 15 '18 at 7:19
$begingroup$
@YangGao The set {0} contains the element 0! Yes, It contains. So what?
$endgroup$
– Rakesh Bhatt
Dec 15 '18 at 9:21
|
show 1 more comment
$begingroup$
Do you think that the set is not closed? If yes then can you give a cluster point which is not in the set? The definition says that A set is closed iff it contains all its cluster point. If 0 is not the cluster point then why do you even worry about it?
$endgroup$
– Rakesh Bhatt
Dec 15 '18 at 6:51
$begingroup$
I got it ! I just thought 0 "should" be a cluster point of the set .Then the set does not have a cluster point, right? Thank you. But wait a minute, the set {0} contains the element 0 !
$endgroup$
– Yang Gao
Dec 15 '18 at 7:06
3
$begingroup$
Post the definition of "cluster point" in your book.
$endgroup$
– xbh
Dec 15 '18 at 7:12
$begingroup$
I think it appears at the answer provided by Anthony Ter .
$endgroup$
– Yang Gao
Dec 15 '18 at 7:19
$begingroup$
@YangGao The set {0} contains the element 0! Yes, It contains. So what?
$endgroup$
– Rakesh Bhatt
Dec 15 '18 at 9:21
$begingroup$
Do you think that the set is not closed? If yes then can you give a cluster point which is not in the set? The definition says that A set is closed iff it contains all its cluster point. If 0 is not the cluster point then why do you even worry about it?
$endgroup$
– Rakesh Bhatt
Dec 15 '18 at 6:51
$begingroup$
Do you think that the set is not closed? If yes then can you give a cluster point which is not in the set? The definition says that A set is closed iff it contains all its cluster point. If 0 is not the cluster point then why do you even worry about it?
$endgroup$
– Rakesh Bhatt
Dec 15 '18 at 6:51
$begingroup$
I got it ! I just thought 0 "should" be a cluster point of the set .Then the set does not have a cluster point, right? Thank you. But wait a minute, the set {0} contains the element 0 !
$endgroup$
– Yang Gao
Dec 15 '18 at 7:06
$begingroup$
I got it ! I just thought 0 "should" be a cluster point of the set .Then the set does not have a cluster point, right? Thank you. But wait a minute, the set {0} contains the element 0 !
$endgroup$
– Yang Gao
Dec 15 '18 at 7:06
3
3
$begingroup$
Post the definition of "cluster point" in your book.
$endgroup$
– xbh
Dec 15 '18 at 7:12
$begingroup$
Post the definition of "cluster point" in your book.
$endgroup$
– xbh
Dec 15 '18 at 7:12
$begingroup$
I think it appears at the answer provided by Anthony Ter .
$endgroup$
– Yang Gao
Dec 15 '18 at 7:19
$begingroup$
I think it appears at the answer provided by Anthony Ter .
$endgroup$
– Yang Gao
Dec 15 '18 at 7:19
$begingroup$
@YangGao The set {0} contains the element 0! Yes, It contains. So what?
$endgroup$
– Rakesh Bhatt
Dec 15 '18 at 9:21
$begingroup$
@YangGao The set {0} contains the element 0! Yes, It contains. So what?
$endgroup$
– Rakesh Bhatt
Dec 15 '18 at 9:21
|
show 1 more comment
1 Answer
1
active
oldest
votes
$begingroup$
If $0$ was a cluster point of ${0}$ then for every $delta > 0$ there would exist an $x neq 0, x in N_delta(0)$ such that $x in {0}$. clearly no such $x$ exists, for any $delta$. Thus $0$ is not a cluster point of ${0}$.
So is ${0 }$ closed? Does it contain all of its limit points? Well, $emptyset subseteq {0}$, so ${0}$ is closed.
answered Dec 15 '18 at 7:04
Anthony TerAnthony Ter
35116
$endgroup$
$begingroup$
Thanks. Now I really get it . The closed set contains all of its cluster point but not all of its elements! Thank you!
$endgroup$
– Yang Gao
Dec 15 '18 at 7:17
$begingroup$
Now the empty set is the only cluster point of the set, right? Just a little "weird".
$endgroup$
– Yang Gao
Dec 15 '18 at 7:25
1
$begingroup$
No,the empty set isn't a point. The empty set represents the set of limit points, and I was trying to show that that set is contained in ${0}$. I should have used $subseteq$, not $in$. I'll edit it now.
$endgroup$
– Anthony Ter
Dec 15 '18 at 7:27
1
$begingroup$
So the set does not have a cluster point ? wow
$endgroup$
– Yang Gao
Dec 15 '18 at 7:31
$begingroup$
Yep, in fact, no finite subset of $mathbb{R}$ has a cluster point.
$endgroup$
– Anthony Ter
Dec 15 '18 at 7:35
|
show 3 more comments
Your Answer
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1 Answer
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$begingroup$
If $0$ was a cluster point of ${0}$ then for every $delta > 0$ there would exist an $x neq 0, x in N_delta(0)$ such that $x in {0}$. clearly no such $x$ exists, for any $delta$. Thus $0$ is not a cluster point of ${0}$.
So is ${0 }$ closed? Does it contain all of its limit points? Well, $emptyset subseteq {0}$, so ${0}$ is closed.
answered Dec 15 '18 at 7:04
Anthony TerAnthony Ter
35116
$endgroup$
$begingroup$
Thanks. Now I really get it . The closed set contains all of its cluster point but not all of its elements! Thank you!
$endgroup$
– Yang Gao
Dec 15 '18 at 7:17
$begingroup$
Now the empty set is the only cluster point of the set, right? Just a little "weird".
$endgroup$
– Yang Gao
Dec 15 '18 at 7:25
1
$begingroup$
No,the empty set isn't a point. The empty set represents the set of limit points, and I was trying to show that that set is contained in ${0}$. I should have used $subseteq$, not $in$. I'll edit it now.
$endgroup$
– Anthony Ter
Dec 15 '18 at 7:27
1
$begingroup$
So the set does not have a cluster point ? wow
$endgroup$
– Yang Gao
Dec 15 '18 at 7:31
$begingroup$
Yep, in fact, no finite subset of $mathbb{R}$ has a cluster point.
$endgroup$
– Anthony Ter
Dec 15 '18 at 7:35
|
show 3 more comments
$begingroup$
If $0$ was a cluster point of ${0}$ then for every $delta > 0$ there would exist an $x neq 0, x in N_delta(0)$ such that $x in {0}$. clearly no such $x$ exists, for any $delta$. Thus $0$ is not a cluster point of ${0}$.
So is ${0 }$ closed? Does it contain all of its limit points? Well, $emptyset subseteq {0}$, so ${0}$ is closed.
answered Dec 15 '18 at 7:04
Anthony TerAnthony Ter
35116
$endgroup$
$begingroup$
Thanks. Now I really get it . The closed set contains all of its cluster point but not all of its elements! Thank you!
$endgroup$
– Yang Gao
Dec 15 '18 at 7:17
$begingroup$
Now the empty set is the only cluster point of the set, right? Just a little "weird".
$endgroup$
– Yang Gao
Dec 15 '18 at 7:25
1
$begingroup$
No,the empty set isn't a point. The empty set represents the set of limit points, and I was trying to show that that set is contained in ${0}$. I should have used $subseteq$, not $in$. I'll edit it now.
$endgroup$
– Anthony Ter
Dec 15 '18 at 7:27
1
$begingroup$
So the set does not have a cluster point ? wow
$endgroup$
– Yang Gao
Dec 15 '18 at 7:31
$begingroup$
Yep, in fact, no finite subset of $mathbb{R}$ has a cluster point.
$endgroup$
– Anthony Ter
Dec 15 '18 at 7:35
|
show 3 more comments
$begingroup$
If $0$ was a cluster point of ${0}$ then for every $delta > 0$ there would exist an $x neq 0, x in N_delta(0)$ such that $x in {0}$. clearly no such $x$ exists, for any $delta$. Thus $0$ is not a cluster point of ${0}$.
So is ${0 }$ closed? Does it contain all of its limit points? Well, $emptyset subseteq {0}$, so ${0}$ is closed.
answered Dec 15 '18 at 7:04
Anthony TerAnthony Ter
35116
$endgroup$
If $0$ was a cluster point of ${0}$ then for every $delta > 0$ there would exist an $x neq 0, x in N_delta(0)$ such that $x in {0}$. clearly no such $x$ exists, for any $delta$. Thus $0$ is not a cluster point of ${0}$.
So is ${0 }$ closed? Does it contain all of its limit points? Well, $emptyset subseteq {0}$, so ${0}$ is closed.
answered Dec 15 '18 at 7:04
Anthony TerAnthony Ter
35116
edited Dec 15 '18 at 7:28
answered Dec 15 '18 at 7:04
Anthony TerAnthony Ter
35116
answered Dec 15 '18 at 7:04
Anthony TerAnthony Ter
35116
answered Dec 15 '18 at 7:04
Anthony TerAnthony Ter
35116
35116
$begingroup$
Thanks. Now I really get it . The closed set contains all of its cluster point but not all of its elements! Thank you!
$endgroup$
– Yang Gao
Dec 15 '18 at 7:17
$begingroup$
Now the empty set is the only cluster point of the set, right? Just a little "weird".
$endgroup$
– Yang Gao
Dec 15 '18 at 7:25
1
$begingroup$
No,the empty set isn't a point. The empty set represents the set of limit points, and I was trying to show that that set is contained in ${0}$. I should have used $subseteq$, not $in$. I'll edit it now.
$endgroup$
– Anthony Ter
Dec 15 '18 at 7:27
1
$begingroup$
So the set does not have a cluster point ? wow
$endgroup$
– Yang Gao
Dec 15 '18 at 7:31
$begingroup$
Yep, in fact, no finite subset of $mathbb{R}$ has a cluster point.
$endgroup$
– Anthony Ter
Dec 15 '18 at 7:35
|
show 3 more comments
$begingroup$
Thanks. Now I really get it . The closed set contains all of its cluster point but not all of its elements! Thank you!
$endgroup$
– Yang Gao
Dec 15 '18 at 7:17
$begingroup$
Now the empty set is the only cluster point of the set, right? Just a little "weird".
$endgroup$
– Yang Gao
Dec 15 '18 at 7:25
1
$begingroup$
No,the empty set isn't a point. The empty set represents the set of limit points, and I was trying to show that that set is contained in ${0}$. I should have used $subseteq$, not $in$. I'll edit it now.
$endgroup$
– Anthony Ter
Dec 15 '18 at 7:27
1
$begingroup$
So the set does not have a cluster point ? wow
$endgroup$
– Yang Gao
Dec 15 '18 at 7:31
$begingroup$
Yep, in fact, no finite subset of $mathbb{R}$ has a cluster point.
$endgroup$
– Anthony Ter
Dec 15 '18 at 7:35
$begingroup$
Thanks. Now I really get it . The closed set contains all of its cluster point but not all of its elements! Thank you!
$endgroup$
– Yang Gao
Dec 15 '18 at 7:17
$begingroup$
Thanks. Now I really get it . The closed set contains all of its cluster point but not all of its elements! Thank you!
$endgroup$
– Yang Gao
Dec 15 '18 at 7:17
$begingroup$
Now the empty set is the only cluster point of the set, right? Just a little "weird".
$endgroup$
– Yang Gao
Dec 15 '18 at 7:25
$begingroup$
Now the empty set is the only cluster point of the set, right? Just a little "weird".
$endgroup$
– Yang Gao
Dec 15 '18 at 7:25
1
1
$begingroup$
No,the empty set isn't a point. The empty set represents the set of limit points, and I was trying to show that that set is contained in ${0}$. I should have used $subseteq$, not $in$. I'll edit it now.
$endgroup$
– Anthony Ter
Dec 15 '18 at 7:27
$begingroup$
No,the empty set isn't a point. The empty set represents the set of limit points, and I was trying to show that that set is contained in ${0}$. I should have used $subseteq$, not $in$. I'll edit it now.
$endgroup$
– Anthony Ter
Dec 15 '18 at 7:27
1
1
$begingroup$
So the set does not have a cluster point ? wow
$endgroup$
– Yang Gao
Dec 15 '18 at 7:31
$begingroup$
So the set does not have a cluster point ? wow
$endgroup$
– Yang Gao
Dec 15 '18 at 7:31
$begingroup$
Yep, in fact, no finite subset of $mathbb{R}$ has a cluster point.
$endgroup$
– Anthony Ter
Dec 15 '18 at 7:35
$begingroup$
Yep, in fact, no finite subset of $mathbb{R}$ has a cluster point.
$endgroup$
– Anthony Ter
Dec 15 '18 at 7:35
|
show 3 more comments
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$begingroup$
Do you think that the set is not closed? If yes then can you give a cluster point which is not in the set? The definition says that A set is closed iff it contains all its cluster point. If 0 is not the cluster point then why do you even worry about it?
$endgroup$
– Rakesh Bhatt
Dec 15 '18 at 6:51
$begingroup$
I got it ! I just thought 0 "should" be a cluster point of the set .Then the set does not have a cluster point, right? Thank you. But wait a minute, the set {0} contains the element 0 !
$endgroup$
– Yang Gao
Dec 15 '18 at 7:06
3
$begingroup$
Post the definition of "cluster point" in your book.
$endgroup$
– xbh
Dec 15 '18 at 7:12
$begingroup$
I think it appears at the answer provided by Anthony Ter .
$endgroup$
– Yang Gao
Dec 15 '18 at 7:19
$begingroup$
@YangGao The set {0} contains the element 0! Yes, It contains. So what?
$endgroup$
– Rakesh Bhatt
Dec 15 '18 at 9:21