Enumerable and Dense
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I'm starting in topology! Something I'm not finding is the relation between enumerability and density, my question is, "A set being dense implies not being enumerable"?
general-topology
asked Jan 1 at 12:38
ValdigleisValdigleis
62
$endgroup$
add a comment |
$begingroup$
I'm starting in topology! Something I'm not finding is the relation between enumerability and density, my question is, "A set being dense implies not being enumerable"?
general-topology
asked Jan 1 at 12:38
ValdigleisValdigleis
62
$endgroup$
2
$begingroup$
Famously $Bbb Q$ is a dense countable subset of $Bbb R$ (in its usual topology).
$endgroup$
– Lord Shark the Unknown
Jan 1 at 12:40
1
$begingroup$
Example of dense and not enumerable is $mathbb Rsetminusmathbb Q$ in $mathbb R$. A dense set can be both enumerable and non-enumerable.
$endgroup$
– Arjun Banerjee
Jan 1 at 12:59
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If X is a topological space then X is a dense subset of X regardless of whether X is countable. The density d(X) is the least infinite cardinal K such that X has a dense subset whose cardinal is at most K. If X does not have a finite dense subset then d(X) is the least cardinal K such that X has a dense subset of cardinality K. Density is one of many topological cardinal functions: Cardinals associated with a topology
$endgroup$
– DanielWainfleet
Jan 1 at 20:28
add a comment |
$begingroup$
I'm starting in topology! Something I'm not finding is the relation between enumerability and density, my question is, "A set being dense implies not being enumerable"?
general-topology
asked Jan 1 at 12:38
ValdigleisValdigleis
62
$endgroup$
I'm starting in topology! Something I'm not finding is the relation between enumerability and density, my question is, "A set being dense implies not being enumerable"?
general-topology
general-topology
asked Jan 1 at 12:38
ValdigleisValdigleis
62
asked Jan 1 at 12:38
ValdigleisValdigleis
62
asked Jan 1 at 12:38
ValdigleisValdigleis
62
asked Jan 1 at 12:38
ValdigleisValdigleis
62
asked Jan 1 at 12:38
ValdigleisValdigleis
62
62
2
$begingroup$
Famously $Bbb Q$ is a dense countable subset of $Bbb R$ (in its usual topology).
$endgroup$
– Lord Shark the Unknown
Jan 1 at 12:40
1
$begingroup$
Example of dense and not enumerable is $mathbb Rsetminusmathbb Q$ in $mathbb R$. A dense set can be both enumerable and non-enumerable.
$endgroup$
– Arjun Banerjee
Jan 1 at 12:59
$begingroup$
If X is a topological space then X is a dense subset of X regardless of whether X is countable. The density d(X) is the least infinite cardinal K such that X has a dense subset whose cardinal is at most K. If X does not have a finite dense subset then d(X) is the least cardinal K such that X has a dense subset of cardinality K. Density is one of many topological cardinal functions: Cardinals associated with a topology
$endgroup$
– DanielWainfleet
Jan 1 at 20:28
add a comment |
2
$begingroup$
Famously $Bbb Q$ is a dense countable subset of $Bbb R$ (in its usual topology).
$endgroup$
– Lord Shark the Unknown
Jan 1 at 12:40
1
$begingroup$
Example of dense and not enumerable is $mathbb Rsetminusmathbb Q$ in $mathbb R$. A dense set can be both enumerable and non-enumerable.
$endgroup$
– Arjun Banerjee
Jan 1 at 12:59
$begingroup$
If X is a topological space then X is a dense subset of X regardless of whether X is countable. The density d(X) is the least infinite cardinal K such that X has a dense subset whose cardinal is at most K. If X does not have a finite dense subset then d(X) is the least cardinal K such that X has a dense subset of cardinality K. Density is one of many topological cardinal functions: Cardinals associated with a topology
$endgroup$
– DanielWainfleet
Jan 1 at 20:28
2
2
$begingroup$
Famously $Bbb Q$ is a dense countable subset of $Bbb R$ (in its usual topology).
$endgroup$
– Lord Shark the Unknown
Jan 1 at 12:40
$begingroup$
Famously $Bbb Q$ is a dense countable subset of $Bbb R$ (in its usual topology).
$endgroup$
– Lord Shark the Unknown
Jan 1 at 12:40
1
1
$begingroup$
Example of dense and not enumerable is $mathbb Rsetminusmathbb Q$ in $mathbb R$. A dense set can be both enumerable and non-enumerable.
$endgroup$
– Arjun Banerjee
Jan 1 at 12:59
$begingroup$
Example of dense and not enumerable is $mathbb Rsetminusmathbb Q$ in $mathbb R$. A dense set can be both enumerable and non-enumerable.
$endgroup$
– Arjun Banerjee
Jan 1 at 12:59
$begingroup$
If X is a topological space then X is a dense subset of X regardless of whether X is countable. The density d(X) is the least infinite cardinal K such that X has a dense subset whose cardinal is at most K. If X does not have a finite dense subset then d(X) is the least cardinal K such that X has a dense subset of cardinality K. Density is one of many topological cardinal functions: Cardinals associated with a topology
$endgroup$
– DanielWainfleet
Jan 1 at 20:28
$begingroup$
If X is a topological space then X is a dense subset of X regardless of whether X is countable. The density d(X) is the least infinite cardinal K such that X has a dense subset whose cardinal is at most K. If X does not have a finite dense subset then d(X) is the least cardinal K such that X has a dense subset of cardinality K. Density is one of many topological cardinal functions: Cardinals associated with a topology
$endgroup$
– DanielWainfleet
Jan 1 at 20:28
add a comment |
1 Answer
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$begingroup$
Density does not imply that a set is enumerable or uncountable. A subset $E$ of a topological space $X$ is dense if every $x in X$ is either in $E$ or is a limit point of $E$ (equivalently, if $overline{E} = X$). The most common example you'll see of this is that $mathbb{Q}$ (which happens to be enumerable) is dense in $mathbb{R}$. However, $mathbb{Q} cup [0, 1]$ (with $[0, 1] subset mathbb{R}$) is dense in $mathbb{R}$ but uncountable.
answered Jan 1 at 14:24
DuncanDuncan
386212
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$begingroup$
Density does not imply that a set is enumerable or uncountable. A subset $E$ of a topological space $X$ is dense if every $x in X$ is either in $E$ or is a limit point of $E$ (equivalently, if $overline{E} = X$). The most common example you'll see of this is that $mathbb{Q}$ (which happens to be enumerable) is dense in $mathbb{R}$. However, $mathbb{Q} cup [0, 1]$ (with $[0, 1] subset mathbb{R}$) is dense in $mathbb{R}$ but uncountable.
answered Jan 1 at 14:24
DuncanDuncan
386212
$endgroup$
add a comment |
$begingroup$
Density does not imply that a set is enumerable or uncountable. A subset $E$ of a topological space $X$ is dense if every $x in X$ is either in $E$ or is a limit point of $E$ (equivalently, if $overline{E} = X$). The most common example you'll see of this is that $mathbb{Q}$ (which happens to be enumerable) is dense in $mathbb{R}$. However, $mathbb{Q} cup [0, 1]$ (with $[0, 1] subset mathbb{R}$) is dense in $mathbb{R}$ but uncountable.
answered Jan 1 at 14:24
DuncanDuncan
386212
$endgroup$
add a comment |
$begingroup$
Density does not imply that a set is enumerable or uncountable. A subset $E$ of a topological space $X$ is dense if every $x in X$ is either in $E$ or is a limit point of $E$ (equivalently, if $overline{E} = X$). The most common example you'll see of this is that $mathbb{Q}$ (which happens to be enumerable) is dense in $mathbb{R}$. However, $mathbb{Q} cup [0, 1]$ (with $[0, 1] subset mathbb{R}$) is dense in $mathbb{R}$ but uncountable.
answered Jan 1 at 14:24
DuncanDuncan
386212
$endgroup$
Density does not imply that a set is enumerable or uncountable. A subset $E$ of a topological space $X$ is dense if every $x in X$ is either in $E$ or is a limit point of $E$ (equivalently, if $overline{E} = X$). The most common example you'll see of this is that $mathbb{Q}$ (which happens to be enumerable) is dense in $mathbb{R}$. However, $mathbb{Q} cup [0, 1]$ (with $[0, 1] subset mathbb{R}$) is dense in $mathbb{R}$ but uncountable.
answered Jan 1 at 14:24
DuncanDuncan
386212
answered Jan 1 at 14:24
DuncanDuncan
386212
answered Jan 1 at 14:24
DuncanDuncan
386212
answered Jan 1 at 14:24
DuncanDuncan
386212
386212
add a comment |
add a comment |
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$begingroup$
Famously $Bbb Q$ is a dense countable subset of $Bbb R$ (in its usual topology).
$endgroup$
– Lord Shark the Unknown
Jan 1 at 12:40
1
$begingroup$
Example of dense and not enumerable is $mathbb Rsetminusmathbb Q$ in $mathbb R$. A dense set can be both enumerable and non-enumerable.
$endgroup$
– Arjun Banerjee
Jan 1 at 12:59
$begingroup$
If X is a topological space then X is a dense subset of X regardless of whether X is countable. The density d(X) is the least infinite cardinal K such that X has a dense subset whose cardinal is at most K. If X does not have a finite dense subset then d(X) is the least cardinal K such that X has a dense subset of cardinality K. Density is one of many topological cardinal functions: Cardinals associated with a topology
$endgroup$
– DanielWainfleet
Jan 1 at 20:28