About residual sets and dense sets
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Suppose $Esubset X$ is some subset of a (complete) metric space $X$. To each $pin E$ we associate a small ball $B(p,epsilon_p)$ centered at $p$, and we want to consider the union
$$
hat E:=bigcup_{pin E} B(p,epsilon_p)
$$
Question: When $E$ is only a dense subset, then $hat E$ may not be the total space. A counter example is that $E=mathbb Q$ and $X=mathbb R$. If we further require $E$ is residual, can we conclude $hat E=X$? If not, any counter example?
general-topology metric-spaces
edited Nov 18 at 21:18
Kelvin Lois
3,0872823
asked Nov 18 at 21:09
Hang
440314
add a comment |
up vote
2
down vote
favorite
Suppose $Esubset X$ is some subset of a (complete) metric space $X$. To each $pin E$ we associate a small ball $B(p,epsilon_p)$ centered at $p$, and we want to consider the union
$$
hat E:=bigcup_{pin E} B(p,epsilon_p)
$$
Question: When $E$ is only a dense subset, then $hat E$ may not be the total space. A counter example is that $E=mathbb Q$ and $X=mathbb R$. If we further require $E$ is residual, can we conclude $hat E=X$? If not, any counter example?
general-topology metric-spaces
edited Nov 18 at 21:18
Kelvin Lois
3,0872823
asked Nov 18 at 21:09
Hang
440314
+1 A very interesting question! I've worked quite a bit with various strengthenings of being a residual set, as well as with various density notions, and the type of union you're forming is not all that different from things I've dealt with, but off-hand I don't know the answer. I suppose that if I'm overlooking something obvious, or at least something I should have thought of, then I'll find out soon enough!
– Dave L. Renfro
Nov 18 at 21:20
3
Suppose that $x in X setminus E$. I bet you can choose each $varepsilon_p$ so that $B(p,varepsilon_p)$ misses $x$.
– Mark McClure
Nov 18 at 22:11
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Suppose $Esubset X$ is some subset of a (complete) metric space $X$. To each $pin E$ we associate a small ball $B(p,epsilon_p)$ centered at $p$, and we want to consider the union
$$
hat E:=bigcup_{pin E} B(p,epsilon_p)
$$
Question: When $E$ is only a dense subset, then $hat E$ may not be the total space. A counter example is that $E=mathbb Q$ and $X=mathbb R$. If we further require $E$ is residual, can we conclude $hat E=X$? If not, any counter example?
general-topology metric-spaces
edited Nov 18 at 21:18
Kelvin Lois
3,0872823
asked Nov 18 at 21:09
Hang
440314
Suppose $Esubset X$ is some subset of a (complete) metric space $X$. To each $pin E$ we associate a small ball $B(p,epsilon_p)$ centered at $p$, and we want to consider the union
$$
hat E:=bigcup_{pin E} B(p,epsilon_p)
$$
Question: When $E$ is only a dense subset, then $hat E$ may not be the total space. A counter example is that $E=mathbb Q$ and $X=mathbb R$. If we further require $E$ is residual, can we conclude $hat E=X$? If not, any counter example?
general-topology metric-spaces
general-topology metric-spaces
edited Nov 18 at 21:18
Kelvin Lois
3,0872823
asked Nov 18 at 21:09
Hang
440314
edited Nov 18 at 21:18
Kelvin Lois
3,0872823
asked Nov 18 at 21:09
Hang
440314
edited Nov 18 at 21:18
Kelvin Lois
3,0872823
edited Nov 18 at 21:18
Kelvin Lois
3,0872823
edited Nov 18 at 21:18
Kelvin Lois
3,0872823
3,0872823
asked Nov 18 at 21:09
Hang
440314
asked Nov 18 at 21:09
Hang
440314
asked Nov 18 at 21:09
Hang
440314
440314
+1 A very interesting question! I've worked quite a bit with various strengthenings of being a residual set, as well as with various density notions, and the type of union you're forming is not all that different from things I've dealt with, but off-hand I don't know the answer. I suppose that if I'm overlooking something obvious, or at least something I should have thought of, then I'll find out soon enough!
– Dave L. Renfro
Nov 18 at 21:20
3
Suppose that $x in X setminus E$. I bet you can choose each $varepsilon_p$ so that $B(p,varepsilon_p)$ misses $x$.
– Mark McClure
Nov 18 at 22:11
add a comment |
+1 A very interesting question! I've worked quite a bit with various strengthenings of being a residual set, as well as with various density notions, and the type of union you're forming is not all that different from things I've dealt with, but off-hand I don't know the answer. I suppose that if I'm overlooking something obvious, or at least something I should have thought of, then I'll find out soon enough!
– Dave L. Renfro
Nov 18 at 21:20
3
Suppose that $x in X setminus E$. I bet you can choose each $varepsilon_p$ so that $B(p,varepsilon_p)$ misses $x$.
– Mark McClure
Nov 18 at 22:11
+1 A very interesting question! I've worked quite a bit with various strengthenings of being a residual set, as well as with various density notions, and the type of union you're forming is not all that different from things I've dealt with, but off-hand I don't know the answer. I suppose that if I'm overlooking something obvious, or at least something I should have thought of, then I'll find out soon enough!
– Dave L. Renfro
Nov 18 at 21:20
+1 A very interesting question! I've worked quite a bit with various strengthenings of being a residual set, as well as with various density notions, and the type of union you're forming is not all that different from things I've dealt with, but off-hand I don't know the answer. I suppose that if I'm overlooking something obvious, or at least something I should have thought of, then I'll find out soon enough!
– Dave L. Renfro
Nov 18 at 21:20
3
3
Suppose that $x in X setminus E$. I bet you can choose each $varepsilon_p$ so that $B(p,varepsilon_p)$ misses $x$.
– Mark McClure
Nov 18 at 22:11
Suppose that $x in X setminus E$. I bet you can choose each $varepsilon_p$ so that $B(p,varepsilon_p)$ misses $x$.
– Mark McClure
Nov 18 at 22:11
add a comment |
1 Answer
1
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oldest
votes
up vote
1
down vote
E.g. if $E$ is the set of irrationals in the reals (which is residual), we can take $r_p =|p|$ for
all irrational $p$ and then $0 notin hat{E}$ for that those radii.
So there is no general guarantee that $hat{E} = X$; we can always avoid specific points of the complement if we so desire.
answered Nov 18 at 22:34
Henno Brandsma
102k345109
This is an "overlooking something obvious" thing that I thought might happen, but now the natural problem arises of determining some upper limits (cardinality, density, etc.) that exist when $E$ is, say, $c$-dense in the space or has full measure or is residual. It's easy to get any isolated set for certain choices of such "big" sets. What about any scattered set, or any countable set, etc.
– Dave L. Renfro
Nov 18 at 22:48
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
E.g. if $E$ is the set of irrationals in the reals (which is residual), we can take $r_p =|p|$ for
all irrational $p$ and then $0 notin hat{E}$ for that those radii.
So there is no general guarantee that $hat{E} = X$; we can always avoid specific points of the complement if we so desire.
answered Nov 18 at 22:34
Henno Brandsma
102k345109
This is an "overlooking something obvious" thing that I thought might happen, but now the natural problem arises of determining some upper limits (cardinality, density, etc.) that exist when $E$ is, say, $c$-dense in the space or has full measure or is residual. It's easy to get any isolated set for certain choices of such "big" sets. What about any scattered set, or any countable set, etc.
– Dave L. Renfro
Nov 18 at 22:48
add a comment |
up vote
1
down vote
E.g. if $E$ is the set of irrationals in the reals (which is residual), we can take $r_p =|p|$ for
all irrational $p$ and then $0 notin hat{E}$ for that those radii.
So there is no general guarantee that $hat{E} = X$; we can always avoid specific points of the complement if we so desire.
answered Nov 18 at 22:34
Henno Brandsma
102k345109
This is an "overlooking something obvious" thing that I thought might happen, but now the natural problem arises of determining some upper limits (cardinality, density, etc.) that exist when $E$ is, say, $c$-dense in the space or has full measure or is residual. It's easy to get any isolated set for certain choices of such "big" sets. What about any scattered set, or any countable set, etc.
– Dave L. Renfro
Nov 18 at 22:48
add a comment |
up vote
1
down vote
up vote
1
down vote
E.g. if $E$ is the set of irrationals in the reals (which is residual), we can take $r_p =|p|$ for
all irrational $p$ and then $0 notin hat{E}$ for that those radii.
So there is no general guarantee that $hat{E} = X$; we can always avoid specific points of the complement if we so desire.
answered Nov 18 at 22:34
Henno Brandsma
102k345109
E.g. if $E$ is the set of irrationals in the reals (which is residual), we can take $r_p =|p|$ for
all irrational $p$ and then $0 notin hat{E}$ for that those radii.
So there is no general guarantee that $hat{E} = X$; we can always avoid specific points of the complement if we so desire.
answered Nov 18 at 22:34
Henno Brandsma
102k345109
answered Nov 18 at 22:34
Henno Brandsma
102k345109
answered Nov 18 at 22:34
Henno Brandsma
102k345109
answered Nov 18 at 22:34
Henno Brandsma
102k345109
102k345109
This is an "overlooking something obvious" thing that I thought might happen, but now the natural problem arises of determining some upper limits (cardinality, density, etc.) that exist when $E$ is, say, $c$-dense in the space or has full measure or is residual. It's easy to get any isolated set for certain choices of such "big" sets. What about any scattered set, or any countable set, etc.
– Dave L. Renfro
Nov 18 at 22:48
add a comment |
This is an "overlooking something obvious" thing that I thought might happen, but now the natural problem arises of determining some upper limits (cardinality, density, etc.) that exist when $E$ is, say, $c$-dense in the space or has full measure or is residual. It's easy to get any isolated set for certain choices of such "big" sets. What about any scattered set, or any countable set, etc.
– Dave L. Renfro
Nov 18 at 22:48
This is an "overlooking something obvious" thing that I thought might happen, but now the natural problem arises of determining some upper limits (cardinality, density, etc.) that exist when $E$ is, say, $c$-dense in the space or has full measure or is residual. It's easy to get any isolated set for certain choices of such "big" sets. What about any scattered set, or any countable set, etc.
– Dave L. Renfro
Nov 18 at 22:48
This is an "overlooking something obvious" thing that I thought might happen, but now the natural problem arises of determining some upper limits (cardinality, density, etc.) that exist when $E$ is, say, $c$-dense in the space or has full measure or is residual. It's easy to get any isolated set for certain choices of such "big" sets. What about any scattered set, or any countable set, etc.
– Dave L. Renfro
Nov 18 at 22:48
add a comment |
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+1 A very interesting question! I've worked quite a bit with various strengthenings of being a residual set, as well as with various density notions, and the type of union you're forming is not all that different from things I've dealt with, but off-hand I don't know the answer. I suppose that if I'm overlooking something obvious, or at least something I should have thought of, then I'll find out soon enough!
– Dave L. Renfro
Nov 18 at 21:20
3
Suppose that $x in X setminus E$. I bet you can choose each $varepsilon_p$ so that $B(p,varepsilon_p)$ misses $x$.
– Mark McClure
Nov 18 at 22:11