Mixing
General topology, compact sets, neighborhood
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I'm really struggling in writing the proof of these statements
My answers are:
1)true
2)true
3)false
but i can't supply the proof for this. Any help please? I don't know how to write proofs. I'm new to this course
real-analysis general-topology compactness
asked Feb 1 at 16:34
PBCPBC
14
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add a comment |
$begingroup$
I'm really struggling in writing the proof of these statements
My answers are:
1)true
2)true
3)false
but i can't supply the proof for this. Any help please? I don't know how to write proofs. I'm new to this course
real-analysis general-topology compactness
asked Feb 1 at 16:34
PBCPBC
14
$endgroup$
$begingroup$
You must provide further context concerning the topological space that is involved. If there is none then think of indiscrete topology for counterexamples.
$endgroup$
– drhab
Feb 1 at 16:41
$begingroup$
This is how the question is asked. But we've been recently working with metric spaces if that's helpful
$endgroup$
– PBC
Feb 1 at 16:43
add a comment |
$begingroup$
I'm really struggling in writing the proof of these statements
My answers are:
1)true
2)true
3)false
but i can't supply the proof for this. Any help please? I don't know how to write proofs. I'm new to this course
real-analysis general-topology compactness
asked Feb 1 at 16:34
PBCPBC
14
$endgroup$
I'm really struggling in writing the proof of these statements
My answers are:
1)true
2)true
3)false
but i can't supply the proof for this. Any help please? I don't know how to write proofs. I'm new to this course
real-analysis general-topology compactness
real-analysis general-topology compactness
asked Feb 1 at 16:34
PBCPBC
14
asked Feb 1 at 16:34
PBCPBC
14
asked Feb 1 at 16:34
PBCPBC
14
asked Feb 1 at 16:34
PBCPBC
14
asked Feb 1 at 16:34
PBCPBC
14
14
$begingroup$
You must provide further context concerning the topological space that is involved. If there is none then think of indiscrete topology for counterexamples.
$endgroup$
– drhab
Feb 1 at 16:41
$begingroup$
This is how the question is asked. But we've been recently working with metric spaces if that's helpful
$endgroup$
– PBC
Feb 1 at 16:43
add a comment |
$begingroup$
You must provide further context concerning the topological space that is involved. If there is none then think of indiscrete topology for counterexamples.
$endgroup$
– drhab
Feb 1 at 16:41
$begingroup$
This is how the question is asked. But we've been recently working with metric spaces if that's helpful
$endgroup$
– PBC
Feb 1 at 16:43
$begingroup$
You must provide further context concerning the topological space that is involved. If there is none then think of indiscrete topology for counterexamples.
$endgroup$
– drhab
Feb 1 at 16:41
$begingroup$
You must provide further context concerning the topological space that is involved. If there is none then think of indiscrete topology for counterexamples.
$endgroup$
– drhab
Feb 1 at 16:41
$begingroup$
This is how the question is asked. But we've been recently working with metric spaces if that's helpful
$endgroup$
– PBC
Feb 1 at 16:43
$begingroup$
This is how the question is asked. But we've been recently working with metric spaces if that's helpful
$endgroup$
– PBC
Feb 1 at 16:43
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
By metric spaces your answers are correct.
1) If $pneq q$ then $d(p,q)>0$ and for $epsilon=frac12d(p,q)>0$ the sets $U_p={xin Xmid d(p,x)<epsilon}$ and $U_q={xin Xmid d(q,x)<epsilon}$ are disjoint open sets containing $p$ and $q$ respectively.
2) Similarly for $i=1,dots,k$ let $U_i$ denote an open set containing $p_i$ and let $V_i$ denote an open set containing $q$ with $U_icap V_i=varnothing$. Now take the union of the $bigcup_{i=1}^nU_i$ and $V=bigcap_{i=1}^n V_i$.
3)Counterexample: $mathbb R$ with usual topology with $p_i=frac1{i}$ and $q=0$.
answered Feb 1 at 16:55
drhabdrhab
104k545136
$endgroup$
$begingroup$
Thank you for your clarification. I didn't understand your answer in part 2) can you explain furthermore?
$endgroup$
– PBC
Feb 1 at 16:59
$begingroup$
In my answer on 2) there was a mistake (repaired now). Sorry for causing confusion.
$endgroup$
– drhab
Feb 1 at 18:31
$begingroup$
What happens then in 2) if we take the union of the ⋃ni=1Ui and V=⋂ni=1Vi
$endgroup$
– PBC
Feb 1 at 18:52
$begingroup$
Both are open, and they are disjoint. The union contains the $p_i $ and the intersection contains $q $.
$endgroup$
– drhab
Feb 1 at 19:41
add a comment |
Your Answer
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1 Answer
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$begingroup$
By metric spaces your answers are correct.
1) If $pneq q$ then $d(p,q)>0$ and for $epsilon=frac12d(p,q)>0$ the sets $U_p={xin Xmid d(p,x)<epsilon}$ and $U_q={xin Xmid d(q,x)<epsilon}$ are disjoint open sets containing $p$ and $q$ respectively.
2) Similarly for $i=1,dots,k$ let $U_i$ denote an open set containing $p_i$ and let $V_i$ denote an open set containing $q$ with $U_icap V_i=varnothing$. Now take the union of the $bigcup_{i=1}^nU_i$ and $V=bigcap_{i=1}^n V_i$.
3)Counterexample: $mathbb R$ with usual topology with $p_i=frac1{i}$ and $q=0$.
answered Feb 1 at 16:55
drhabdrhab
104k545136
$endgroup$
$begingroup$
Thank you for your clarification. I didn't understand your answer in part 2) can you explain furthermore?
$endgroup$
– PBC
Feb 1 at 16:59
$begingroup$
In my answer on 2) there was a mistake (repaired now). Sorry for causing confusion.
$endgroup$
– drhab
Feb 1 at 18:31
$begingroup$
What happens then in 2) if we take the union of the ⋃ni=1Ui and V=⋂ni=1Vi
$endgroup$
– PBC
Feb 1 at 18:52
$begingroup$
Both are open, and they are disjoint. The union contains the $p_i $ and the intersection contains $q $.
$endgroup$
– drhab
Feb 1 at 19:41
add a comment |
$begingroup$
By metric spaces your answers are correct.
1) If $pneq q$ then $d(p,q)>0$ and for $epsilon=frac12d(p,q)>0$ the sets $U_p={xin Xmid d(p,x)<epsilon}$ and $U_q={xin Xmid d(q,x)<epsilon}$ are disjoint open sets containing $p$ and $q$ respectively.
2) Similarly for $i=1,dots,k$ let $U_i$ denote an open set containing $p_i$ and let $V_i$ denote an open set containing $q$ with $U_icap V_i=varnothing$. Now take the union of the $bigcup_{i=1}^nU_i$ and $V=bigcap_{i=1}^n V_i$.
3)Counterexample: $mathbb R$ with usual topology with $p_i=frac1{i}$ and $q=0$.
answered Feb 1 at 16:55
drhabdrhab
104k545136
$endgroup$
$begingroup$
Thank you for your clarification. I didn't understand your answer in part 2) can you explain furthermore?
$endgroup$
– PBC
Feb 1 at 16:59
$begingroup$
In my answer on 2) there was a mistake (repaired now). Sorry for causing confusion.
$endgroup$
– drhab
Feb 1 at 18:31
$begingroup$
What happens then in 2) if we take the union of the ⋃ni=1Ui and V=⋂ni=1Vi
$endgroup$
– PBC
Feb 1 at 18:52
$begingroup$
Both are open, and they are disjoint. The union contains the $p_i $ and the intersection contains $q $.
$endgroup$
– drhab
Feb 1 at 19:41
add a comment |
$begingroup$
By metric spaces your answers are correct.
1) If $pneq q$ then $d(p,q)>0$ and for $epsilon=frac12d(p,q)>0$ the sets $U_p={xin Xmid d(p,x)<epsilon}$ and $U_q={xin Xmid d(q,x)<epsilon}$ are disjoint open sets containing $p$ and $q$ respectively.
2) Similarly for $i=1,dots,k$ let $U_i$ denote an open set containing $p_i$ and let $V_i$ denote an open set containing $q$ with $U_icap V_i=varnothing$. Now take the union of the $bigcup_{i=1}^nU_i$ and $V=bigcap_{i=1}^n V_i$.
3)Counterexample: $mathbb R$ with usual topology with $p_i=frac1{i}$ and $q=0$.
answered Feb 1 at 16:55
drhabdrhab
104k545136
$endgroup$
By metric spaces your answers are correct.
1) If $pneq q$ then $d(p,q)>0$ and for $epsilon=frac12d(p,q)>0$ the sets $U_p={xin Xmid d(p,x)<epsilon}$ and $U_q={xin Xmid d(q,x)<epsilon}$ are disjoint open sets containing $p$ and $q$ respectively.
2) Similarly for $i=1,dots,k$ let $U_i$ denote an open set containing $p_i$ and let $V_i$ denote an open set containing $q$ with $U_icap V_i=varnothing$. Now take the union of the $bigcup_{i=1}^nU_i$ and $V=bigcap_{i=1}^n V_i$.
3)Counterexample: $mathbb R$ with usual topology with $p_i=frac1{i}$ and $q=0$.
answered Feb 1 at 16:55
drhabdrhab
104k545136
edited Feb 1 at 18:30
answered Feb 1 at 16:55
drhabdrhab
104k545136
answered Feb 1 at 16:55
drhabdrhab
104k545136
answered Feb 1 at 16:55
drhabdrhab
104k545136
104k545136
$begingroup$
Thank you for your clarification. I didn't understand your answer in part 2) can you explain furthermore?
$endgroup$
– PBC
Feb 1 at 16:59
$begingroup$
In my answer on 2) there was a mistake (repaired now). Sorry for causing confusion.
$endgroup$
– drhab
Feb 1 at 18:31
$begingroup$
What happens then in 2) if we take the union of the ⋃ni=1Ui and V=⋂ni=1Vi
$endgroup$
– PBC
Feb 1 at 18:52
$begingroup$
Both are open, and they are disjoint. The union contains the $p_i $ and the intersection contains $q $.
$endgroup$
– drhab
Feb 1 at 19:41
add a comment |
$begingroup$
Thank you for your clarification. I didn't understand your answer in part 2) can you explain furthermore?
$endgroup$
– PBC
Feb 1 at 16:59
$begingroup$
In my answer on 2) there was a mistake (repaired now). Sorry for causing confusion.
$endgroup$
– drhab
Feb 1 at 18:31
$begingroup$
What happens then in 2) if we take the union of the ⋃ni=1Ui and V=⋂ni=1Vi
$endgroup$
– PBC
Feb 1 at 18:52
$begingroup$
Both are open, and they are disjoint. The union contains the $p_i $ and the intersection contains $q $.
$endgroup$
– drhab
Feb 1 at 19:41
$begingroup$
Thank you for your clarification. I didn't understand your answer in part 2) can you explain furthermore?
$endgroup$
– PBC
Feb 1 at 16:59
$begingroup$
Thank you for your clarification. I didn't understand your answer in part 2) can you explain furthermore?
$endgroup$
– PBC
Feb 1 at 16:59
$begingroup$
In my answer on 2) there was a mistake (repaired now). Sorry for causing confusion.
$endgroup$
– drhab
Feb 1 at 18:31
$begingroup$
In my answer on 2) there was a mistake (repaired now). Sorry for causing confusion.
$endgroup$
– drhab
Feb 1 at 18:31
$begingroup$
What happens then in 2) if we take the union of the ⋃ni=1Ui and V=⋂ni=1Vi
$endgroup$
– PBC
Feb 1 at 18:52
$begingroup$
What happens then in 2) if we take the union of the ⋃ni=1Ui and V=⋂ni=1Vi
$endgroup$
– PBC
Feb 1 at 18:52
$begingroup$
Both are open, and they are disjoint. The union contains the $p_i $ and the intersection contains $q $.
$endgroup$
– drhab
Feb 1 at 19:41
$begingroup$
Both are open, and they are disjoint. The union contains the $p_i $ and the intersection contains $q $.
$endgroup$
– drhab
Feb 1 at 19:41
add a comment |
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$begingroup$
You must provide further context concerning the topological space that is involved. If there is none then think of indiscrete topology for counterexamples.
$endgroup$
– drhab
Feb 1 at 16:41
$begingroup$
This is how the question is asked. But we've been recently working with metric spaces if that's helpful
$endgroup$
– PBC
Feb 1 at 16:43