Mixing
Neigborhood of a point
$begingroup$
I have a homework that says the following:
If $N_delta(p)$ is a neighborhood of $p$ that does not intersect $p,$ show that it cannot intersect $E'$.
where E is a subset of R and where the set made by all the accumulation points of E is called the derivative set and it is indicated with E′
Any help in this proof would be appreciated.Thank you
this is my attempt
general-topology
asked Jan 31 at 17:49
PBCPBC
14
$endgroup$
|
show 8 more comments
$begingroup$
I have a homework that says the following:
If $N_delta(p)$ is a neighborhood of $p$ that does not intersect $p,$ show that it cannot intersect $E'$.
where E is a subset of R and where the set made by all the accumulation points of E is called the derivative set and it is indicated with E′
Any help in this proof would be appreciated.Thank you
this is my attempt
general-topology
asked Jan 31 at 17:49
PBCPBC
14
$endgroup$
$begingroup$
Please tell us what $E'$ denotes.
$endgroup$
– md2perpe
Jan 31 at 18:23
$begingroup$
@md2perpe The set made by all the accumulation points of E is called the derivative set and it is indicated with E′
$endgroup$
– PBC
Jan 31 at 18:24
$begingroup$
So, then, what set is $E$? You just refer to it without having defined it.
$endgroup$
– md2perpe
Jan 31 at 18:25
$begingroup$
@md2perpe E is a subset of R
$endgroup$
– PBC
Jan 31 at 18:29
1
$begingroup$
Exercise 6 talks about $E$ but I don't think that $E$ in exercise 7 has to be the same as in exercise 6. However, I think that "does not intersect $p$" should be "does not intersect $E$", since that makes more sense. Thus, I think that the exercise should read "Let $E subseteq mathbb{R}$ and $p in mathbb{R}$. Assume that $N_delta(p)$ is an open neighborhood of $p$ that does not intersect $E$. Show that it also cannot intersect $E'$."
$endgroup$
– md2perpe
Jan 31 at 20:09
|
show 8 more comments
$begingroup$
I have a homework that says the following:
If $N_delta(p)$ is a neighborhood of $p$ that does not intersect $p,$ show that it cannot intersect $E'$.
where E is a subset of R and where the set made by all the accumulation points of E is called the derivative set and it is indicated with E′
Any help in this proof would be appreciated.Thank you
this is my attempt
general-topology
asked Jan 31 at 17:49
PBCPBC
14
$endgroup$
I have a homework that says the following:
If $N_delta(p)$ is a neighborhood of $p$ that does not intersect $p,$ show that it cannot intersect $E'$.
where E is a subset of R and where the set made by all the accumulation points of E is called the derivative set and it is indicated with E′
Any help in this proof would be appreciated.Thank you
this is my attempt
general-topology
general-topology
asked Jan 31 at 17:49
PBCPBC
14
asked Jan 31 at 17:49
PBCPBC
14
edited Jan 31 at 21:49
PBC
asked Jan 31 at 17:49
PBCPBC
14
asked Jan 31 at 17:49
PBCPBC
14
asked Jan 31 at 17:49
PBCPBC
14
14
$begingroup$
Please tell us what $E'$ denotes.
$endgroup$
– md2perpe
Jan 31 at 18:23
$begingroup$
@md2perpe The set made by all the accumulation points of E is called the derivative set and it is indicated with E′
$endgroup$
– PBC
Jan 31 at 18:24
$begingroup$
So, then, what set is $E$? You just refer to it without having defined it.
$endgroup$
– md2perpe
Jan 31 at 18:25
$begingroup$
@md2perpe E is a subset of R
$endgroup$
– PBC
Jan 31 at 18:29
1
$begingroup$
Exercise 6 talks about $E$ but I don't think that $E$ in exercise 7 has to be the same as in exercise 6. However, I think that "does not intersect $p$" should be "does not intersect $E$", since that makes more sense. Thus, I think that the exercise should read "Let $E subseteq mathbb{R}$ and $p in mathbb{R}$. Assume that $N_delta(p)$ is an open neighborhood of $p$ that does not intersect $E$. Show that it also cannot intersect $E'$."
$endgroup$
– md2perpe
Jan 31 at 20:09
|
show 8 more comments
$begingroup$
Please tell us what $E'$ denotes.
$endgroup$
– md2perpe
Jan 31 at 18:23
$begingroup$
@md2perpe The set made by all the accumulation points of E is called the derivative set and it is indicated with E′
$endgroup$
– PBC
Jan 31 at 18:24
$begingroup$
So, then, what set is $E$? You just refer to it without having defined it.
$endgroup$
– md2perpe
Jan 31 at 18:25
$begingroup$
@md2perpe E is a subset of R
$endgroup$
– PBC
Jan 31 at 18:29
1
$begingroup$
Exercise 6 talks about $E$ but I don't think that $E$ in exercise 7 has to be the same as in exercise 6. However, I think that "does not intersect $p$" should be "does not intersect $E$", since that makes more sense. Thus, I think that the exercise should read "Let $E subseteq mathbb{R}$ and $p in mathbb{R}$. Assume that $N_delta(p)$ is an open neighborhood of $p$ that does not intersect $E$. Show that it also cannot intersect $E'$."
$endgroup$
– md2perpe
Jan 31 at 20:09
$begingroup$
Please tell us what $E'$ denotes.
$endgroup$
– md2perpe
Jan 31 at 18:23
$begingroup$
Please tell us what $E'$ denotes.
$endgroup$
– md2perpe
Jan 31 at 18:23
$begingroup$
@md2perpe The set made by all the accumulation points of E is called the derivative set and it is indicated with E′
$endgroup$
– PBC
Jan 31 at 18:24
$begingroup$
@md2perpe The set made by all the accumulation points of E is called the derivative set and it is indicated with E′
$endgroup$
– PBC
Jan 31 at 18:24
$begingroup$
So, then, what set is $E$? You just refer to it without having defined it.
$endgroup$
– md2perpe
Jan 31 at 18:25
$begingroup$
So, then, what set is $E$? You just refer to it without having defined it.
$endgroup$
– md2perpe
Jan 31 at 18:25
$begingroup$
@md2perpe E is a subset of R
$endgroup$
– PBC
Jan 31 at 18:29
$begingroup$
@md2perpe E is a subset of R
$endgroup$
– PBC
Jan 31 at 18:29
1
1
$begingroup$
Exercise 6 talks about $E$ but I don't think that $E$ in exercise 7 has to be the same as in exercise 6. However, I think that "does not intersect $p$" should be "does not intersect $E$", since that makes more sense. Thus, I think that the exercise should read "Let $E subseteq mathbb{R}$ and $p in mathbb{R}$. Assume that $N_delta(p)$ is an open neighborhood of $p$ that does not intersect $E$. Show that it also cannot intersect $E'$."
$endgroup$
– md2perpe
Jan 31 at 20:09
$begingroup$
Exercise 6 talks about $E$ but I don't think that $E$ in exercise 7 has to be the same as in exercise 6. However, I think that "does not intersect $p$" should be "does not intersect $E$", since that makes more sense. Thus, I think that the exercise should read "Let $E subseteq mathbb{R}$ and $p in mathbb{R}$. Assume that $N_delta(p)$ is an open neighborhood of $p$ that does not intersect $E$. Show that it also cannot intersect $E'$."
$endgroup$
– md2perpe
Jan 31 at 20:09
|
show 8 more comments
1 Answer
1
active
oldest
votes
$begingroup$
I think there is a typo in your problem set. I believe this supposed to read "Let $N_delta(p)$ be a neighborhood of $p$ that doesn't intersect $E$. Show that it cannot intersect $E'$."
We'll prove the contrapositive: Suppose $N_delta(p) cap E' neq emptyset$. So there is some $x in N_delta(p)$, such that $x$ is a limit point of $E$. Let $epsilon < delta - d(x,p)$. Since $x$ is a limit point of $E$, there is some $e in E$ such that $e in N_epsilon(x)$. Can you take it from here?
WHY I THINK THIS IS THE CORRECT INTERPRETATION:
OP looks to be taking a first course in real analysis, which grounds the level of problem difficulty to be expected. Any neighborhood of $p$ must intersect $p$ (as I interpret this), so the problem must lie there. Note that this must mean $p not in E$, which would explain why the problem doesn't state $p in E$.
answered Jan 31 at 20:08
JoeJoe
75129
$endgroup$
$begingroup$
yes this my first course in real analysis and yes, there might be a typo in it, so take it the way you did. I'm struggling a bit with it. If you please can provide a proof for this proposition. That would be much appreciated.
$endgroup$
– PBC
Jan 31 at 20:25
$begingroup$
Why don't you write out your thoughts and I'll help you along the way.
$endgroup$
– Joe
Jan 31 at 20:25
$begingroup$
That's the problem, this is my first class for the neighborhood thing, so i don't have any thoughts about how to prove it. If you can solve it please, then i'll look at your steps and understand how it's done.
$endgroup$
– PBC
Jan 31 at 20:28
$begingroup$
Here's a hint: it will be easier to prove the contrapositive. That is, let $N_delta(p)$ intersect $E'$. Show that it also intersects $E$. Start by using the definition of the derived set, $E'$...
$endgroup$
– Joe
Jan 31 at 20:31
$begingroup$
Can you show me how to do it please?
$endgroup$
– PBC
Jan 31 at 20:35
|
show 13 more comments
Your Answer
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1 Answer
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1 Answer
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oldest
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votes
$begingroup$
I think there is a typo in your problem set. I believe this supposed to read "Let $N_delta(p)$ be a neighborhood of $p$ that doesn't intersect $E$. Show that it cannot intersect $E'$."
We'll prove the contrapositive: Suppose $N_delta(p) cap E' neq emptyset$. So there is some $x in N_delta(p)$, such that $x$ is a limit point of $E$. Let $epsilon < delta - d(x,p)$. Since $x$ is a limit point of $E$, there is some $e in E$ such that $e in N_epsilon(x)$. Can you take it from here?
WHY I THINK THIS IS THE CORRECT INTERPRETATION:
OP looks to be taking a first course in real analysis, which grounds the level of problem difficulty to be expected. Any neighborhood of $p$ must intersect $p$ (as I interpret this), so the problem must lie there. Note that this must mean $p not in E$, which would explain why the problem doesn't state $p in E$.
answered Jan 31 at 20:08
JoeJoe
75129
$endgroup$
$begingroup$
yes this my first course in real analysis and yes, there might be a typo in it, so take it the way you did. I'm struggling a bit with it. If you please can provide a proof for this proposition. That would be much appreciated.
$endgroup$
– PBC
Jan 31 at 20:25
$begingroup$
Why don't you write out your thoughts and I'll help you along the way.
$endgroup$
– Joe
Jan 31 at 20:25
$begingroup$
That's the problem, this is my first class for the neighborhood thing, so i don't have any thoughts about how to prove it. If you can solve it please, then i'll look at your steps and understand how it's done.
$endgroup$
– PBC
Jan 31 at 20:28
$begingroup$
Here's a hint: it will be easier to prove the contrapositive. That is, let $N_delta(p)$ intersect $E'$. Show that it also intersects $E$. Start by using the definition of the derived set, $E'$...
$endgroup$
– Joe
Jan 31 at 20:31
$begingroup$
Can you show me how to do it please?
$endgroup$
– PBC
Jan 31 at 20:35
|
show 13 more comments
$begingroup$
I think there is a typo in your problem set. I believe this supposed to read "Let $N_delta(p)$ be a neighborhood of $p$ that doesn't intersect $E$. Show that it cannot intersect $E'$."
We'll prove the contrapositive: Suppose $N_delta(p) cap E' neq emptyset$. So there is some $x in N_delta(p)$, such that $x$ is a limit point of $E$. Let $epsilon < delta - d(x,p)$. Since $x$ is a limit point of $E$, there is some $e in E$ such that $e in N_epsilon(x)$. Can you take it from here?
WHY I THINK THIS IS THE CORRECT INTERPRETATION:
OP looks to be taking a first course in real analysis, which grounds the level of problem difficulty to be expected. Any neighborhood of $p$ must intersect $p$ (as I interpret this), so the problem must lie there. Note that this must mean $p not in E$, which would explain why the problem doesn't state $p in E$.
answered Jan 31 at 20:08
JoeJoe
75129
$endgroup$
$begingroup$
yes this my first course in real analysis and yes, there might be a typo in it, so take it the way you did. I'm struggling a bit with it. If you please can provide a proof for this proposition. That would be much appreciated.
$endgroup$
– PBC
Jan 31 at 20:25
$begingroup$
Why don't you write out your thoughts and I'll help you along the way.
$endgroup$
– Joe
Jan 31 at 20:25
$begingroup$
That's the problem, this is my first class for the neighborhood thing, so i don't have any thoughts about how to prove it. If you can solve it please, then i'll look at your steps and understand how it's done.
$endgroup$
– PBC
Jan 31 at 20:28
$begingroup$
Here's a hint: it will be easier to prove the contrapositive. That is, let $N_delta(p)$ intersect $E'$. Show that it also intersects $E$. Start by using the definition of the derived set, $E'$...
$endgroup$
– Joe
Jan 31 at 20:31
$begingroup$
Can you show me how to do it please?
$endgroup$
– PBC
Jan 31 at 20:35
|
show 13 more comments
$begingroup$
I think there is a typo in your problem set. I believe this supposed to read "Let $N_delta(p)$ be a neighborhood of $p$ that doesn't intersect $E$. Show that it cannot intersect $E'$."
We'll prove the contrapositive: Suppose $N_delta(p) cap E' neq emptyset$. So there is some $x in N_delta(p)$, such that $x$ is a limit point of $E$. Let $epsilon < delta - d(x,p)$. Since $x$ is a limit point of $E$, there is some $e in E$ such that $e in N_epsilon(x)$. Can you take it from here?
WHY I THINK THIS IS THE CORRECT INTERPRETATION:
OP looks to be taking a first course in real analysis, which grounds the level of problem difficulty to be expected. Any neighborhood of $p$ must intersect $p$ (as I interpret this), so the problem must lie there. Note that this must mean $p not in E$, which would explain why the problem doesn't state $p in E$.
answered Jan 31 at 20:08
JoeJoe
75129
$endgroup$
I think there is a typo in your problem set. I believe this supposed to read "Let $N_delta(p)$ be a neighborhood of $p$ that doesn't intersect $E$. Show that it cannot intersect $E'$."
We'll prove the contrapositive: Suppose $N_delta(p) cap E' neq emptyset$. So there is some $x in N_delta(p)$, such that $x$ is a limit point of $E$. Let $epsilon < delta - d(x,p)$. Since $x$ is a limit point of $E$, there is some $e in E$ such that $e in N_epsilon(x)$. Can you take it from here?
WHY I THINK THIS IS THE CORRECT INTERPRETATION:
OP looks to be taking a first course in real analysis, which grounds the level of problem difficulty to be expected. Any neighborhood of $p$ must intersect $p$ (as I interpret this), so the problem must lie there. Note that this must mean $p not in E$, which would explain why the problem doesn't state $p in E$.
answered Jan 31 at 20:08
JoeJoe
75129
edited Feb 1 at 19:17
answered Jan 31 at 20:08
JoeJoe
75129
answered Jan 31 at 20:08
JoeJoe
75129
answered Jan 31 at 20:08
JoeJoe
75129
75129
$begingroup$
yes this my first course in real analysis and yes, there might be a typo in it, so take it the way you did. I'm struggling a bit with it. If you please can provide a proof for this proposition. That would be much appreciated.
$endgroup$
– PBC
Jan 31 at 20:25
$begingroup$
Why don't you write out your thoughts and I'll help you along the way.
$endgroup$
– Joe
Jan 31 at 20:25
$begingroup$
That's the problem, this is my first class for the neighborhood thing, so i don't have any thoughts about how to prove it. If you can solve it please, then i'll look at your steps and understand how it's done.
$endgroup$
– PBC
Jan 31 at 20:28
$begingroup$
Here's a hint: it will be easier to prove the contrapositive. That is, let $N_delta(p)$ intersect $E'$. Show that it also intersects $E$. Start by using the definition of the derived set, $E'$...
$endgroup$
– Joe
Jan 31 at 20:31
$begingroup$
Can you show me how to do it please?
$endgroup$
– PBC
Jan 31 at 20:35
|
show 13 more comments
$begingroup$
yes this my first course in real analysis and yes, there might be a typo in it, so take it the way you did. I'm struggling a bit with it. If you please can provide a proof for this proposition. That would be much appreciated.
$endgroup$
– PBC
Jan 31 at 20:25
$begingroup$
Why don't you write out your thoughts and I'll help you along the way.
$endgroup$
– Joe
Jan 31 at 20:25
$begingroup$
That's the problem, this is my first class for the neighborhood thing, so i don't have any thoughts about how to prove it. If you can solve it please, then i'll look at your steps and understand how it's done.
$endgroup$
– PBC
Jan 31 at 20:28
$begingroup$
Here's a hint: it will be easier to prove the contrapositive. That is, let $N_delta(p)$ intersect $E'$. Show that it also intersects $E$. Start by using the definition of the derived set, $E'$...
$endgroup$
– Joe
Jan 31 at 20:31
$begingroup$
Can you show me how to do it please?
$endgroup$
– PBC
Jan 31 at 20:35
$begingroup$
yes this my first course in real analysis and yes, there might be a typo in it, so take it the way you did. I'm struggling a bit with it. If you please can provide a proof for this proposition. That would be much appreciated.
$endgroup$
– PBC
Jan 31 at 20:25
$begingroup$
yes this my first course in real analysis and yes, there might be a typo in it, so take it the way you did. I'm struggling a bit with it. If you please can provide a proof for this proposition. That would be much appreciated.
$endgroup$
– PBC
Jan 31 at 20:25
$begingroup$
Why don't you write out your thoughts and I'll help you along the way.
$endgroup$
– Joe
Jan 31 at 20:25
$begingroup$
Why don't you write out your thoughts and I'll help you along the way.
$endgroup$
– Joe
Jan 31 at 20:25
$begingroup$
That's the problem, this is my first class for the neighborhood thing, so i don't have any thoughts about how to prove it. If you can solve it please, then i'll look at your steps and understand how it's done.
$endgroup$
– PBC
Jan 31 at 20:28
$begingroup$
That's the problem, this is my first class for the neighborhood thing, so i don't have any thoughts about how to prove it. If you can solve it please, then i'll look at your steps and understand how it's done.
$endgroup$
– PBC
Jan 31 at 20:28
$begingroup$
Here's a hint: it will be easier to prove the contrapositive. That is, let $N_delta(p)$ intersect $E'$. Show that it also intersects $E$. Start by using the definition of the derived set, $E'$...
$endgroup$
– Joe
Jan 31 at 20:31
$begingroup$
Here's a hint: it will be easier to prove the contrapositive. That is, let $N_delta(p)$ intersect $E'$. Show that it also intersects $E$. Start by using the definition of the derived set, $E'$...
$endgroup$
– Joe
Jan 31 at 20:31
$begingroup$
Can you show me how to do it please?
$endgroup$
– PBC
Jan 31 at 20:35
$begingroup$
Can you show me how to do it please?
$endgroup$
– PBC
Jan 31 at 20:35
|
show 13 more comments
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$begingroup$
Please tell us what $E'$ denotes.
$endgroup$
– md2perpe
Jan 31 at 18:23
$begingroup$
@md2perpe The set made by all the accumulation points of E is called the derivative set and it is indicated with E′
$endgroup$
– PBC
Jan 31 at 18:24
$begingroup$
So, then, what set is $E$? You just refer to it without having defined it.
$endgroup$
– md2perpe
Jan 31 at 18:25
$begingroup$
@md2perpe E is a subset of R
$endgroup$
– PBC
Jan 31 at 18:29
1
$begingroup$
Exercise 6 talks about $E$ but I don't think that $E$ in exercise 7 has to be the same as in exercise 6. However, I think that "does not intersect $p$" should be "does not intersect $E$", since that makes more sense. Thus, I think that the exercise should read "Let $E subseteq mathbb{R}$ and $p in mathbb{R}$. Assume that $N_delta(p)$ is an open neighborhood of $p$ that does not intersect $E$. Show that it also cannot intersect $E'$."
$endgroup$
– md2perpe
Jan 31 at 20:09