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"?










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  • 2




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    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






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    Example of dense and not enumerable is $mathbb Rsetminusmathbb Q$ in $mathbb R$. A dense set can be both enumerable and non-enumerable.
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    – 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


















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$begingroup$


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"?










share|cite|improve this question









$endgroup$








  • 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












  • $begingroup$
    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
















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$begingroup$


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"?










share|cite|improve this question









$endgroup$




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






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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
















  • 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












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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.






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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.






    share|cite|improve this answer









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      0












      $begingroup$

      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.






      share|cite|improve this answer









      $endgroup$
















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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.






        share|cite|improve this answer









        $endgroup$



        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.







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        answered Jan 1 at 14:24









        DuncanDuncan

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