Nontrivial example of closed set relative to subspace but not with respect to original space
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I am interested in finding the example of the set which is close relative subspace but not in original space.
I know that $mathbb Q $ is the subspace of $mathbb R$ And $mathbb Q$ is close relative to itself but not with respect to $mathbb R$.
But I am interested in the nontrivial example not like above I had mentioned.
Any Help will be appreciated
general-topology analysis metric-spaces
asked Aug 14 at 7:57
SRJ
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up vote
1
down vote
favorite
I am interested in finding the example of the set which is close relative subspace but not in original space.
I know that $mathbb Q $ is the subspace of $mathbb R$ And $mathbb Q$ is close relative to itself but not with respect to $mathbb R$.
But I am interested in the nontrivial example not like above I had mentioned.
Any Help will be appreciated
general-topology analysis metric-spaces
asked Aug 14 at 7:57
SRJ
1,298417
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I am interested in finding the example of the set which is close relative subspace but not in original space.
I know that $mathbb Q $ is the subspace of $mathbb R$ And $mathbb Q$ is close relative to itself but not with respect to $mathbb R$.
But I am interested in the nontrivial example not like above I had mentioned.
Any Help will be appreciated
general-topology analysis metric-spaces
asked Aug 14 at 7:57
SRJ
1,298417
I am interested in finding the example of the set which is close relative subspace but not in original space.
I know that $mathbb Q $ is the subspace of $mathbb R$ And $mathbb Q$ is close relative to itself but not with respect to $mathbb R$.
But I am interested in the nontrivial example not like above I had mentioned.
Any Help will be appreciated
general-topology analysis metric-spaces
asked Aug 14 at 7:57
SRJ
1,298417
asked Aug 14 at 7:57
SRJ
1,298417
asked Aug 14 at 7:57
SRJ
1,298417
asked Aug 14 at 7:57
SRJ
1,298417
1,298417
add a comment |Â
add a comment |Â
1 Answer
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This really depends on your notion of trivial. The interval $(0,1]$ is closed in $(0,2)$, for example. Another example would be the open unit ball in $mathbbR^2$ considered on the subspace of the latter, $B_1(0,0) cup B_1(2,0)$. Or for example, if you take $X = bigcup_i in mathbbZ L_i$ with $L_i = (0,1) times i$, then each $L_i$ is closed in $X$.
More generally, $F$ is closed in a subspace $Y$ of a space $X$ iff it is of the form $F = C cap Y$ with $C$ closed in $X$. This gives a way of generating examples, as non-trivial as you can come up with.
answered Aug 14 at 8:09
Guido A.
4,101725
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
This really depends on your notion of trivial. The interval $(0,1]$ is closed in $(0,2)$, for example. Another example would be the open unit ball in $mathbbR^2$ considered on the subspace of the latter, $B_1(0,0) cup B_1(2,0)$. Or for example, if you take $X = bigcup_i in mathbbZ L_i$ with $L_i = (0,1) times i$, then each $L_i$ is closed in $X$.
More generally, $F$ is closed in a subspace $Y$ of a space $X$ iff it is of the form $F = C cap Y$ with $C$ closed in $X$. This gives a way of generating examples, as non-trivial as you can come up with.
answered Aug 14 at 8:09
Guido A.
4,101725
add a comment |Â
up vote
2
down vote
accepted
This really depends on your notion of trivial. The interval $(0,1]$ is closed in $(0,2)$, for example. Another example would be the open unit ball in $mathbbR^2$ considered on the subspace of the latter, $B_1(0,0) cup B_1(2,0)$. Or for example, if you take $X = bigcup_i in mathbbZ L_i$ with $L_i = (0,1) times i$, then each $L_i$ is closed in $X$.
More generally, $F$ is closed in a subspace $Y$ of a space $X$ iff it is of the form $F = C cap Y$ with $C$ closed in $X$. This gives a way of generating examples, as non-trivial as you can come up with.
answered Aug 14 at 8:09
Guido A.
4,101725
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
This really depends on your notion of trivial. The interval $(0,1]$ is closed in $(0,2)$, for example. Another example would be the open unit ball in $mathbbR^2$ considered on the subspace of the latter, $B_1(0,0) cup B_1(2,0)$. Or for example, if you take $X = bigcup_i in mathbbZ L_i$ with $L_i = (0,1) times i$, then each $L_i$ is closed in $X$.
More generally, $F$ is closed in a subspace $Y$ of a space $X$ iff it is of the form $F = C cap Y$ with $C$ closed in $X$. This gives a way of generating examples, as non-trivial as you can come up with.
answered Aug 14 at 8:09
Guido A.
4,101725
This really depends on your notion of trivial. The interval $(0,1]$ is closed in $(0,2)$, for example. Another example would be the open unit ball in $mathbbR^2$ considered on the subspace of the latter, $B_1(0,0) cup B_1(2,0)$. Or for example, if you take $X = bigcup_i in mathbbZ L_i$ with $L_i = (0,1) times i$, then each $L_i$ is closed in $X$.
More generally, $F$ is closed in a subspace $Y$ of a space $X$ iff it is of the form $F = C cap Y$ with $C$ closed in $X$. This gives a way of generating examples, as non-trivial as you can come up with.
answered Aug 14 at 8:09
Guido A.
4,101725
edited Aug 14 at 8:33
answered Aug 14 at 8:09
Guido A.
4,101725
answered Aug 14 at 8:09
Guido A.
4,101725
answered Aug 14 at 8:09
Guido A.
4,101725
4,101725
add a comment |Â
add a comment |Â
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