Prove or disprove if a quotient map from X to Y with Y Hausdorff, then X is Hausdorff. [closed]
$begingroup$
For two open disjoint subsets U and V, I want to show their pre-images are disjoint open subsets of X or not. But I have no idea how to do it.
Any help would be appreciated. Thanks in advance!
quotient-spaces separation-axioms
asked Dec 17 '18 at 0:52
FlashhhFlashhh
325
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closed as off-topic by Nosrati, Saad, user10354138, Ben, naveen dankal Dec 17 '18 at 6:15
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, Saad, user10354138, Ben, naveen dankal
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
For two open disjoint subsets U and V, I want to show their pre-images are disjoint open subsets of X or not. But I have no idea how to do it.
Any help would be appreciated. Thanks in advance!
quotient-spaces separation-axioms
asked Dec 17 '18 at 0:52
FlashhhFlashhh
325
$endgroup$
closed as off-topic by Nosrati, Saad, user10354138, Ben, naveen dankal Dec 17 '18 at 6:15
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, Saad, user10354138, Ben, naveen dankal
If this question can be reworded to fit the rules in the help center, please edit the question.
1
$begingroup$
hint: let $Y$ be the one-point space
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– Matt Booth
Dec 17 '18 at 1:58
add a comment |
$begingroup$
For two open disjoint subsets U and V, I want to show their pre-images are disjoint open subsets of X or not. But I have no idea how to do it.
Any help would be appreciated. Thanks in advance!
quotient-spaces separation-axioms
asked Dec 17 '18 at 0:52
FlashhhFlashhh
325
$endgroup$
For two open disjoint subsets U and V, I want to show their pre-images are disjoint open subsets of X or not. But I have no idea how to do it.
Any help would be appreciated. Thanks in advance!
quotient-spaces separation-axioms
quotient-spaces separation-axioms
asked Dec 17 '18 at 0:52
FlashhhFlashhh
325
asked Dec 17 '18 at 0:52
FlashhhFlashhh
325
asked Dec 17 '18 at 0:52
FlashhhFlashhh
325
asked Dec 17 '18 at 0:52
FlashhhFlashhh
325
asked Dec 17 '18 at 0:52
FlashhhFlashhh
325
325
closed as off-topic by Nosrati, Saad, user10354138, Ben, naveen dankal Dec 17 '18 at 6:15
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, Saad, user10354138, Ben, naveen dankal
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Nosrati, Saad, user10354138, Ben, naveen dankal Dec 17 '18 at 6:15
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, Saad, user10354138, Ben, naveen dankal
If this question can be reworded to fit the rules in the help center, please edit the question.
1
$begingroup$
hint: let $Y$ be the one-point space
$endgroup$
– Matt Booth
Dec 17 '18 at 1:58
add a comment |
1
$begingroup$
hint: let $Y$ be the one-point space
$endgroup$
– Matt Booth
Dec 17 '18 at 1:58
1
1
$begingroup$
hint: let $Y$ be the one-point space
$endgroup$
– Matt Booth
Dec 17 '18 at 1:58
$begingroup$
hint: let $Y$ be the one-point space
$endgroup$
– Matt Booth
Dec 17 '18 at 1:58
add a comment |
1 Answer
1
active
oldest
votes
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This is not true. Map $mathbb R$ with the indiscrete topology to any one point set. The one point set is Hausdorff while $mathbb R$ under the indiscrete topology is not.
answered Dec 17 '18 at 1:34
John DoumaJohn Douma
5,59711419
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1
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It might be useful for some readers to point out that there's nothing special about $mathbb R$ here. Any set with at least two elements would work as well.
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– Andreas Blass
Dec 17 '18 at 3:00
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@AndreasBlass Agreed. I upvoted your comment to draw any reader's attention to it.
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– John Douma
Dec 17 '18 at 4:55
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
This is not true. Map $mathbb R$ with the indiscrete topology to any one point set. The one point set is Hausdorff while $mathbb R$ under the indiscrete topology is not.
answered Dec 17 '18 at 1:34
John DoumaJohn Douma
5,59711419
$endgroup$
1
$begingroup$
It might be useful for some readers to point out that there's nothing special about $mathbb R$ here. Any set with at least two elements would work as well.
$endgroup$
– Andreas Blass
Dec 17 '18 at 3:00
$begingroup$
@AndreasBlass Agreed. I upvoted your comment to draw any reader's attention to it.
$endgroup$
– John Douma
Dec 17 '18 at 4:55
add a comment |
$begingroup$
This is not true. Map $mathbb R$ with the indiscrete topology to any one point set. The one point set is Hausdorff while $mathbb R$ under the indiscrete topology is not.
answered Dec 17 '18 at 1:34
John DoumaJohn Douma
5,59711419
$endgroup$
1
$begingroup$
It might be useful for some readers to point out that there's nothing special about $mathbb R$ here. Any set with at least two elements would work as well.
$endgroup$
– Andreas Blass
Dec 17 '18 at 3:00
$begingroup$
@AndreasBlass Agreed. I upvoted your comment to draw any reader's attention to it.
$endgroup$
– John Douma
Dec 17 '18 at 4:55
add a comment |
$begingroup$
This is not true. Map $mathbb R$ with the indiscrete topology to any one point set. The one point set is Hausdorff while $mathbb R$ under the indiscrete topology is not.
answered Dec 17 '18 at 1:34
John DoumaJohn Douma
5,59711419
$endgroup$
This is not true. Map $mathbb R$ with the indiscrete topology to any one point set. The one point set is Hausdorff while $mathbb R$ under the indiscrete topology is not.
answered Dec 17 '18 at 1:34
John DoumaJohn Douma
5,59711419
edited Dec 17 '18 at 1:57
answered Dec 17 '18 at 1:34
John DoumaJohn Douma
5,59711419
answered Dec 17 '18 at 1:34
John DoumaJohn Douma
5,59711419
answered Dec 17 '18 at 1:34
John DoumaJohn Douma
5,59711419
5,59711419
1
$begingroup$
It might be useful for some readers to point out that there's nothing special about $mathbb R$ here. Any set with at least two elements would work as well.
$endgroup$
– Andreas Blass
Dec 17 '18 at 3:00
$begingroup$
@AndreasBlass Agreed. I upvoted your comment to draw any reader's attention to it.
$endgroup$
– John Douma
Dec 17 '18 at 4:55
add a comment |
1
$begingroup$
It might be useful for some readers to point out that there's nothing special about $mathbb R$ here. Any set with at least two elements would work as well.
$endgroup$
– Andreas Blass
Dec 17 '18 at 3:00
$begingroup$
@AndreasBlass Agreed. I upvoted your comment to draw any reader's attention to it.
$endgroup$
– John Douma
Dec 17 '18 at 4:55
1
1
$begingroup$
It might be useful for some readers to point out that there's nothing special about $mathbb R$ here. Any set with at least two elements would work as well.
$endgroup$
– Andreas Blass
Dec 17 '18 at 3:00
$begingroup$
It might be useful for some readers to point out that there's nothing special about $mathbb R$ here. Any set with at least two elements would work as well.
$endgroup$
– Andreas Blass
Dec 17 '18 at 3:00
$begingroup$
@AndreasBlass Agreed. I upvoted your comment to draw any reader's attention to it.
$endgroup$
– John Douma
Dec 17 '18 at 4:55
$begingroup$
@AndreasBlass Agreed. I upvoted your comment to draw any reader's attention to it.
$endgroup$
– John Douma
Dec 17 '18 at 4:55
add a comment |
1
$begingroup$
hint: let $Y$ be the one-point space
$endgroup$
– Matt Booth
Dec 17 '18 at 1:58