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Give an example of a space which is separable but no proper subspace of it is separable. [closed]
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I need an example of a topological space which is itself separable but no proper subspace of it separable.
general-topology examples-counterexamples
asked Jan 16 at 18:49
Amrita DeyAmrita Dey
113
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closed as off-topic by Alexander Gruber♦ Jan 17 at 2:59
This question appears to be off-topic. The users who voted to close gave this specific reason:
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$begingroup$
I need an example of a topological space which is itself separable but no proper subspace of it separable.
general-topology examples-counterexamples
asked Jan 16 at 18:49
Amrita DeyAmrita Dey
113
$endgroup$
closed as off-topic by Alexander Gruber♦ Jan 17 at 2:59
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." – Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
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Welcome to Mathematics Stack Exchange! A quick tour will hep you understand how best to form questions and answers. The lingua franca for formulation is MathJax.
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– dantopa
Jan 16 at 21:26
add a comment |
$begingroup$
I need an example of a topological space which is itself separable but no proper subspace of it separable.
general-topology examples-counterexamples
asked Jan 16 at 18:49
Amrita DeyAmrita Dey
113
$endgroup$
I need an example of a topological space which is itself separable but no proper subspace of it separable.
general-topology examples-counterexamples
general-topology examples-counterexamples
asked Jan 16 at 18:49
Amrita DeyAmrita Dey
113
asked Jan 16 at 18:49
Amrita DeyAmrita Dey
113
asked Jan 16 at 18:49
Amrita DeyAmrita Dey
113
asked Jan 16 at 18:49
Amrita DeyAmrita Dey
113
asked Jan 16 at 18:49
Amrita DeyAmrita Dey
113
113
closed as off-topic by Alexander Gruber♦ Jan 17 at 2:59
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." – Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Alexander Gruber♦ Jan 17 at 2:59
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." – Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
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Welcome to Mathematics Stack Exchange! A quick tour will hep you understand how best to form questions and answers. The lingua franca for formulation is MathJax.
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– dantopa
Jan 16 at 21:26
add a comment |
$begingroup$
Welcome to Mathematics Stack Exchange! A quick tour will hep you understand how best to form questions and answers. The lingua franca for formulation is MathJax.
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– dantopa
Jan 16 at 21:26
$begingroup$
Welcome to Mathematics Stack Exchange! A quick tour will hep you understand how best to form questions and answers. The lingua franca for formulation is MathJax.
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– dantopa
Jan 16 at 21:26
$begingroup$
Welcome to Mathematics Stack Exchange! A quick tour will hep you understand how best to form questions and answers. The lingua franca for formulation is MathJax.
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– dantopa
Jan 16 at 21:26
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2 Answers
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A singelton ${x}$. It has no proper subspaces!
answered Jan 16 at 18:52
ploosu2ploosu2
4,6431024
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The empty set is a proper subspace. (But, you could instead consider the empty space, which indeed has no proper subspaces.)
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– Eric Wofsey
Jan 16 at 19:22
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Suppose $X$ were such a space as you ask for. As $X$ is separable, there is a (at most) countable dense subset $D$ of $X$. Then clearly $D$ is a separable subspace of $X$. So if your condition holds, it cannot be a proper subset, so $D=X$ and $X$ is a countable space. But then any proper subset is countable and so dense. So then $X$ cannot have any proper subspaces, i.e. $X = emptyset$.
answered Jan 16 at 21:47
Henno BrandsmaHenno Brandsma
110k348118
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add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
A singelton ${x}$. It has no proper subspaces!
answered Jan 16 at 18:52
ploosu2ploosu2
4,6431024
$endgroup$
$begingroup$
The empty set is a proper subspace. (But, you could instead consider the empty space, which indeed has no proper subspaces.)
$endgroup$
– Eric Wofsey
Jan 16 at 19:22
add a comment |
$begingroup$
A singelton ${x}$. It has no proper subspaces!
answered Jan 16 at 18:52
ploosu2ploosu2
4,6431024
$endgroup$
$begingroup$
The empty set is a proper subspace. (But, you could instead consider the empty space, which indeed has no proper subspaces.)
$endgroup$
– Eric Wofsey
Jan 16 at 19:22
add a comment |
$begingroup$
A singelton ${x}$. It has no proper subspaces!
answered Jan 16 at 18:52
ploosu2ploosu2
4,6431024
$endgroup$
A singelton ${x}$. It has no proper subspaces!
answered Jan 16 at 18:52
ploosu2ploosu2
4,6431024
answered Jan 16 at 18:52
ploosu2ploosu2
4,6431024
answered Jan 16 at 18:52
ploosu2ploosu2
4,6431024
answered Jan 16 at 18:52
ploosu2ploosu2
4,6431024
4,6431024
$begingroup$
The empty set is a proper subspace. (But, you could instead consider the empty space, which indeed has no proper subspaces.)
$endgroup$
– Eric Wofsey
Jan 16 at 19:22
add a comment |
$begingroup$
The empty set is a proper subspace. (But, you could instead consider the empty space, which indeed has no proper subspaces.)
$endgroup$
– Eric Wofsey
Jan 16 at 19:22
$begingroup$
The empty set is a proper subspace. (But, you could instead consider the empty space, which indeed has no proper subspaces.)
$endgroup$
– Eric Wofsey
Jan 16 at 19:22
$begingroup$
The empty set is a proper subspace. (But, you could instead consider the empty space, which indeed has no proper subspaces.)
$endgroup$
– Eric Wofsey
Jan 16 at 19:22
add a comment |
$begingroup$
Suppose $X$ were such a space as you ask for. As $X$ is separable, there is a (at most) countable dense subset $D$ of $X$. Then clearly $D$ is a separable subspace of $X$. So if your condition holds, it cannot be a proper subset, so $D=X$ and $X$ is a countable space. But then any proper subset is countable and so dense. So then $X$ cannot have any proper subspaces, i.e. $X = emptyset$.
answered Jan 16 at 21:47
Henno BrandsmaHenno Brandsma
110k348118
$endgroup$
add a comment |
$begingroup$
Suppose $X$ were such a space as you ask for. As $X$ is separable, there is a (at most) countable dense subset $D$ of $X$. Then clearly $D$ is a separable subspace of $X$. So if your condition holds, it cannot be a proper subset, so $D=X$ and $X$ is a countable space. But then any proper subset is countable and so dense. So then $X$ cannot have any proper subspaces, i.e. $X = emptyset$.
answered Jan 16 at 21:47
Henno BrandsmaHenno Brandsma
110k348118
$endgroup$
add a comment |
$begingroup$
Suppose $X$ were such a space as you ask for. As $X$ is separable, there is a (at most) countable dense subset $D$ of $X$. Then clearly $D$ is a separable subspace of $X$. So if your condition holds, it cannot be a proper subset, so $D=X$ and $X$ is a countable space. But then any proper subset is countable and so dense. So then $X$ cannot have any proper subspaces, i.e. $X = emptyset$.
answered Jan 16 at 21:47
Henno BrandsmaHenno Brandsma
110k348118
$endgroup$
Suppose $X$ were such a space as you ask for. As $X$ is separable, there is a (at most) countable dense subset $D$ of $X$. Then clearly $D$ is a separable subspace of $X$. So if your condition holds, it cannot be a proper subset, so $D=X$ and $X$ is a countable space. But then any proper subset is countable and so dense. So then $X$ cannot have any proper subspaces, i.e. $X = emptyset$.
answered Jan 16 at 21:47
Henno BrandsmaHenno Brandsma
110k348118
answered Jan 16 at 21:47
Henno BrandsmaHenno Brandsma
110k348118
answered Jan 16 at 21:47
Henno BrandsmaHenno Brandsma
110k348118
answered Jan 16 at 21:47
Henno BrandsmaHenno Brandsma
110k348118
110k348118
add a comment |
add a comment |
$begingroup$
Welcome to Mathematics Stack Exchange! A quick tour will hep you understand how best to form questions and answers. The lingua franca for formulation is MathJax.
$endgroup$
– dantopa
Jan 16 at 21:26