Logical meaning of a sentence
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There is one question, which says, "Give an example of a separable space $left( X, mathscr{T} right)$ in which there is an uncountable set which does not contain any of its limit point".
Now, the last part, which says, "A set which does not contain any of its limit points" is a bit confusing. Is it equivalent to "No point in the set is a limit point of the set"?
The confusion arises because, maybe the set has no limit points at all (and hence no point of the set is a limit point) and then vacously, it contains all of its limit points (which is exact opposite of the statement).
general-topology logic
asked Nov 27 at 6:47
Aniruddha Deshmukh
789418
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up vote
1
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There is one question, which says, "Give an example of a separable space $left( X, mathscr{T} right)$ in which there is an uncountable set which does not contain any of its limit point".
Now, the last part, which says, "A set which does not contain any of its limit points" is a bit confusing. Is it equivalent to "No point in the set is a limit point of the set"?
The confusion arises because, maybe the set has no limit points at all (and hence no point of the set is a limit point) and then vacously, it contains all of its limit points (which is exact opposite of the statement).
general-topology logic
asked Nov 27 at 6:47
Aniruddha Deshmukh
789418
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
There is one question, which says, "Give an example of a separable space $left( X, mathscr{T} right)$ in which there is an uncountable set which does not contain any of its limit point".
Now, the last part, which says, "A set which does not contain any of its limit points" is a bit confusing. Is it equivalent to "No point in the set is a limit point of the set"?
The confusion arises because, maybe the set has no limit points at all (and hence no point of the set is a limit point) and then vacously, it contains all of its limit points (which is exact opposite of the statement).
general-topology logic
asked Nov 27 at 6:47
Aniruddha Deshmukh
789418
There is one question, which says, "Give an example of a separable space $left( X, mathscr{T} right)$ in which there is an uncountable set which does not contain any of its limit point".
Now, the last part, which says, "A set which does not contain any of its limit points" is a bit confusing. Is it equivalent to "No point in the set is a limit point of the set"?
The confusion arises because, maybe the set has no limit points at all (and hence no point of the set is a limit point) and then vacously, it contains all of its limit points (which is exact opposite of the statement).
general-topology logic
general-topology logic
asked Nov 27 at 6:47
Aniruddha Deshmukh
789418
asked Nov 27 at 6:47
Aniruddha Deshmukh
789418
asked Nov 27 at 6:47
Aniruddha Deshmukh
789418
asked Nov 27 at 6:47
Aniruddha Deshmukh
789418
asked Nov 27 at 6:47
Aniruddha Deshmukh
789418
789418
add a comment |
add a comment |
2 Answers
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3
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If a set has no limit points, then it simultaneously contains all its limit points and none of its limit points. There is no contradiction here, and that's exactly the kind of example I would look for. At least at first. Uncountable is tricky, though, so you may instead have to look for an example which has limit points but doesn't contain them.
answered Nov 27 at 6:57
Arthur
110k7104186
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0
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You are being asked to give an example of a separable space $X$ which has an uncountable subset $Y$ such that any limit point $y$ of $Y$ in $X$ has $y not in Y$. If $Y$ somehow had no limit points, you'd be done.
answered Nov 27 at 6:58
Patrick Stevens
28.2k52874
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
If a set has no limit points, then it simultaneously contains all its limit points and none of its limit points. There is no contradiction here, and that's exactly the kind of example I would look for. At least at first. Uncountable is tricky, though, so you may instead have to look for an example which has limit points but doesn't contain them.
answered Nov 27 at 6:57
Arthur
110k7104186
add a comment |
up vote
3
down vote
accepted
If a set has no limit points, then it simultaneously contains all its limit points and none of its limit points. There is no contradiction here, and that's exactly the kind of example I would look for. At least at first. Uncountable is tricky, though, so you may instead have to look for an example which has limit points but doesn't contain them.
answered Nov 27 at 6:57
Arthur
110k7104186
add a comment |
up vote
3
down vote
accepted
up vote
3
down vote
accepted
If a set has no limit points, then it simultaneously contains all its limit points and none of its limit points. There is no contradiction here, and that's exactly the kind of example I would look for. At least at first. Uncountable is tricky, though, so you may instead have to look for an example which has limit points but doesn't contain them.
answered Nov 27 at 6:57
Arthur
110k7104186
If a set has no limit points, then it simultaneously contains all its limit points and none of its limit points. There is no contradiction here, and that's exactly the kind of example I would look for. At least at first. Uncountable is tricky, though, so you may instead have to look for an example which has limit points but doesn't contain them.
answered Nov 27 at 6:57
Arthur
110k7104186
answered Nov 27 at 6:57
Arthur
110k7104186
answered Nov 27 at 6:57
Arthur
110k7104186
answered Nov 27 at 6:57
Arthur
110k7104186
110k7104186
add a comment |
add a comment |
up vote
0
down vote
You are being asked to give an example of a separable space $X$ which has an uncountable subset $Y$ such that any limit point $y$ of $Y$ in $X$ has $y not in Y$. If $Y$ somehow had no limit points, you'd be done.
answered Nov 27 at 6:58
Patrick Stevens
28.2k52874
add a comment |
up vote
0
down vote
You are being asked to give an example of a separable space $X$ which has an uncountable subset $Y$ such that any limit point $y$ of $Y$ in $X$ has $y not in Y$. If $Y$ somehow had no limit points, you'd be done.
answered Nov 27 at 6:58
Patrick Stevens
28.2k52874
add a comment |
up vote
0
down vote
up vote
0
down vote
You are being asked to give an example of a separable space $X$ which has an uncountable subset $Y$ such that any limit point $y$ of $Y$ in $X$ has $y not in Y$. If $Y$ somehow had no limit points, you'd be done.
answered Nov 27 at 6:58
Patrick Stevens
28.2k52874
You are being asked to give an example of a separable space $X$ which has an uncountable subset $Y$ such that any limit point $y$ of $Y$ in $X$ has $y not in Y$. If $Y$ somehow had no limit points, you'd be done.
answered Nov 27 at 6:58
Patrick Stevens
28.2k52874
answered Nov 27 at 6:58
Patrick Stevens
28.2k52874
answered Nov 27 at 6:58
Patrick Stevens
28.2k52874
answered Nov 27 at 6:58
Patrick Stevens
28.2k52874
28.2k52874
add a comment |
add a comment |
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