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Prove every subset of in the discrete metric is clopen



Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)discrete metric, both open and closed.discrete metric, both open and closed.Open set in a metric space is union of closed balls?Every metric space contains a discrete, coarsely dense subsetProving that a subset endowed with the discrete metric is both open and closed - choice of radius of the ball around a pointIn a metric space, is every open set the countable union of closed sets?Compact Sets of $(X,d)$ with discrete metricDoes there exist any non discrete metric space $(X,d)$ in which every $F_sigma$ (resp. $G_delta$) set is clopen?Open and closed balls in discrete metricOpen and Closed Sets Discrete MetricHow to prove the set of bounded sequences is clopen in the uniform metric?










1












$begingroup$


Hey fellow math enthusiasts! I am reading in ”Introduction to Topology” by Gameline and Greene and I got stuck on an exercise in the first chapter, and I’d love some help on understanding their solution. The problem is as follows:



”Given a set $X$ and metric $d(x, y) = 1$ if $x neq y$ and $d(x, y) = 0$ if $x = y$ then we want to prove that every subset of the resulting metric space $(X, d)$ is both open and closed.”.



And the solution is as follows:



”Since each ball $B(x; frac12)$ reduces to the singleton set $x$, every subset is a union of open balls, hence every subset is open.”.



My interpretation of the solution is that they are just providing the way to reason. They only showed that each subset is open but not closed.



In the book there is a theorem that states that a subset of $X$ is open if and only if it is a union of open balls in $X$, and is being used in the proof.



I get that in $X$ each subset is a singleton set $x$ or a collection of singletons and since each singleton can be rewritten in X as an open ball $B(x; frac12)$ then each collection of singletons can be written as a union of these open balls and thus each subset of $X$ is open.



But how do we get that each subset is closed? My idea is that we look at the complements of the sets we considered above. Since each of these complement-sets also obey the same structure (so in practice would not be distinguishable from the sets above) they too, using the ball-argument, can be showed to be open sets. Then using the argument (theorem in book) that if a subset is open, it’s complement is closed. So both the subsets of individual singletons or collections of singletons are both open and closed.



How do I formalize? Any feedback is greatly appreciated. Thanks in advance.



/Isak






metric-spaces







share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    Possible duplicate of discrete metric, both open and closed.
    $endgroup$
    – YuiTo Cheng
    18 hours ago






  • 1




    $begingroup$
    You essentially have the entire formal argument here. It may be that the original text considered the part you added to be easy enough for the reader to figure out themselves.
    $endgroup$
    – Ben Millwood
    14 hours ago










  • $begingroup$
    @Ben Millwood when you say ”here” in the first sentence, do you mean in my argumentation or in the link YuiTo Cheng supplied?
    $endgroup$
    – iaenstrom
    9 hours ago















1












$begingroup$


Hey fellow math enthusiasts! I am reading in ”Introduction to Topology” by Gameline and Greene and I got stuck on an exercise in the first chapter, and I’d love some help on understanding their solution. The problem is as follows:



”Given a set $X$ and metric $d(x, y) = 1$ if $x neq y$ and $d(x, y) = 0$ if $x = y$ then we want to prove that every subset of the resulting metric space $(X, d)$ is both open and closed.”.



And the solution is as follows:



”Since each ball $B(x; frac12)$ reduces to the singleton set $x$, every subset is a union of open balls, hence every subset is open.”.



My interpretation of the solution is that they are just providing the way to reason. They only showed that each subset is open but not closed.



In the book there is a theorem that states that a subset of $X$ is open if and only if it is a union of open balls in $X$, and is being used in the proof.



I get that in $X$ each subset is a singleton set $x$ or a collection of singletons and since each singleton can be rewritten in X as an open ball $B(x; frac12)$ then each collection of singletons can be written as a union of these open balls and thus each subset of $X$ is open.



But how do we get that each subset is closed? My idea is that we look at the complements of the sets we considered above. Since each of these complement-sets also obey the same structure (so in practice would not be distinguishable from the sets above) they too, using the ball-argument, can be showed to be open sets. Then using the argument (theorem in book) that if a subset is open, it’s complement is closed. So both the subsets of individual singletons or collections of singletons are both open and closed.



How do I formalize? Any feedback is greatly appreciated. Thanks in advance.



/Isak






metric-spaces







share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    Possible duplicate of discrete metric, both open and closed.
    $endgroup$
    – YuiTo Cheng
    18 hours ago






  • 1




    $begingroup$
    You essentially have the entire formal argument here. It may be that the original text considered the part you added to be easy enough for the reader to figure out themselves.
    $endgroup$
    – Ben Millwood
    14 hours ago










  • $begingroup$
    @Ben Millwood when you say ”here” in the first sentence, do you mean in my argumentation or in the link YuiTo Cheng supplied?
    $endgroup$
    – iaenstrom
    9 hours ago













1












1








1





$begingroup$


Hey fellow math enthusiasts! I am reading in ”Introduction to Topology” by Gameline and Greene and I got stuck on an exercise in the first chapter, and I’d love some help on understanding their solution. The problem is as follows:



”Given a set $X$ and metric $d(x, y) = 1$ if $x neq y$ and $d(x, y) = 0$ if $x = y$ then we want to prove that every subset of the resulting metric space $(X, d)$ is both open and closed.”.



And the solution is as follows:



”Since each ball $B(x; frac12)$ reduces to the singleton set $x$, every subset is a union of open balls, hence every subset is open.”.



My interpretation of the solution is that they are just providing the way to reason. They only showed that each subset is open but not closed.



In the book there is a theorem that states that a subset of $X$ is open if and only if it is a union of open balls in $X$, and is being used in the proof.



I get that in $X$ each subset is a singleton set $x$ or a collection of singletons and since each singleton can be rewritten in X as an open ball $B(x; frac12)$ then each collection of singletons can be written as a union of these open balls and thus each subset of $X$ is open.



But how do we get that each subset is closed? My idea is that we look at the complements of the sets we considered above. Since each of these complement-sets also obey the same structure (so in practice would not be distinguishable from the sets above) they too, using the ball-argument, can be showed to be open sets. Then using the argument (theorem in book) that if a subset is open, it’s complement is closed. So both the subsets of individual singletons or collections of singletons are both open and closed.



How do I formalize? Any feedback is greatly appreciated. Thanks in advance.



/Isak






metric-spaces







share|cite|improve this question











$endgroup$




Hey fellow math enthusiasts! I am reading in ”Introduction to Topology” by Gameline and Greene and I got stuck on an exercise in the first chapter, and I’d love some help on understanding their solution. The problem is as follows:



”Given a set $X$ and metric $d(x, y) = 1$ if $x neq y$ and $d(x, y) = 0$ if $x = y$ then we want to prove that every subset of the resulting metric space $(X, d)$ is both open and closed.”.



And the solution is as follows:



”Since each ball $B(x; frac12)$ reduces to the singleton set $x$, every subset is a union of open balls, hence every subset is open.”.



My interpretation of the solution is that they are just providing the way to reason. They only showed that each subset is open but not closed.



In the book there is a theorem that states that a subset of $X$ is open if and only if it is a union of open balls in $X$, and is being used in the proof.



I get that in $X$ each subset is a singleton set $x$ or a collection of singletons and since each singleton can be rewritten in X as an open ball $B(x; frac12)$ then each collection of singletons can be written as a union of these open balls and thus each subset of $X$ is open.



But how do we get that each subset is closed? My idea is that we look at the complements of the sets we considered above. Since each of these complement-sets also obey the same structure (so in practice would not be distinguishable from the sets above) they too, using the ball-argument, can be showed to be open sets. Then using the argument (theorem in book) that if a subset is open, it’s complement is closed. So both the subsets of individual singletons or collections of singletons are both open and closed.



How do I formalize? Any feedback is greatly appreciated. Thanks in advance.



/Isak






metric-spaces




metric-spaces






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 18 hours ago









YuiTo Cheng

2,41341037




2,41341037










asked 19 hours ago









iaenstromiaenstrom

487




487







  • 1




    $begingroup$
    Possible duplicate of discrete metric, both open and closed.
    $endgroup$
    – YuiTo Cheng
    18 hours ago






  • 1




    $begingroup$
    You essentially have the entire formal argument here. It may be that the original text considered the part you added to be easy enough for the reader to figure out themselves.
    $endgroup$
    – Ben Millwood
    14 hours ago










  • $begingroup$
    @Ben Millwood when you say ”here” in the first sentence, do you mean in my argumentation or in the link YuiTo Cheng supplied?
    $endgroup$
    – iaenstrom
    9 hours ago












  • 1




    $begingroup$
    Possible duplicate of discrete metric, both open and closed.
    $endgroup$
    – YuiTo Cheng
    18 hours ago






  • 1




    $begingroup$
    You essentially have the entire formal argument here. It may be that the original text considered the part you added to be easy enough for the reader to figure out themselves.
    $endgroup$
    – Ben Millwood
    14 hours ago










  • $begingroup$
    @Ben Millwood when you say ”here” in the first sentence, do you mean in my argumentation or in the link YuiTo Cheng supplied?
    $endgroup$
    – iaenstrom
    9 hours ago







1




1




$begingroup$
Possible duplicate of discrete metric, both open and closed.
$endgroup$
– YuiTo Cheng
18 hours ago




$begingroup$
Possible duplicate of discrete metric, both open and closed.
$endgroup$
– YuiTo Cheng
18 hours ago




1




1




$begingroup$
You essentially have the entire formal argument here. It may be that the original text considered the part you added to be easy enough for the reader to figure out themselves.
$endgroup$
– Ben Millwood
14 hours ago




$begingroup$
You essentially have the entire formal argument here. It may be that the original text considered the part you added to be easy enough for the reader to figure out themselves.
$endgroup$
– Ben Millwood
14 hours ago












$begingroup$
@Ben Millwood when you say ”here” in the first sentence, do you mean in my argumentation or in the link YuiTo Cheng supplied?
$endgroup$
– iaenstrom
9 hours ago




$begingroup$
@Ben Millwood when you say ”here” in the first sentence, do you mean in my argumentation or in the link YuiTo Cheng supplied?
$endgroup$
– iaenstrom
9 hours ago










2 Answers
2






active

oldest

votes


















6












$begingroup$

Let every subset of a topolgical space be open. A subset is closed if and only if its complement is open. The complement of every subset is a subset so is open. Therefore every subset is closed






share|cite|improve this answer









$endgroup$




















    2












    $begingroup$

    Another way to reason: suppose $x in overlineA$ (the closure of $A$) for some arbitary subset of $A$. Then every ball around $x$ intersects $A$, in particular $B(x,frac12)=x$ must intersect $A$, which means $x in A$.



    So for all $A subseteq X$, $overlineA subseteq A (subseteq overlineA)$ so $A = overlineA$ and every subset $A$ is closed.






    share|cite|improve this answer









    $endgroup$













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      2 Answers
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      2 Answers
      2






      active

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      active

      oldest

      votes






      active

      oldest

      votes









      6












      $begingroup$

      Let every subset of a topolgical space be open. A subset is closed if and only if its complement is open. The complement of every subset is a subset so is open. Therefore every subset is closed






      share|cite|improve this answer









      $endgroup$

















        6












        $begingroup$

        Let every subset of a topolgical space be open. A subset is closed if and only if its complement is open. The complement of every subset is a subset so is open. Therefore every subset is closed






        share|cite|improve this answer









        $endgroup$















          6












          6








          6





          $begingroup$

          Let every subset of a topolgical space be open. A subset is closed if and only if its complement is open. The complement of every subset is a subset so is open. Therefore every subset is closed






          share|cite|improve this answer









          $endgroup$



          Let every subset of a topolgical space be open. A subset is closed if and only if its complement is open. The complement of every subset is a subset so is open. Therefore every subset is closed







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 19 hours ago









          G AkerG Aker

          3037




          3037





















              2












              $begingroup$

              Another way to reason: suppose $x in overlineA$ (the closure of $A$) for some arbitary subset of $A$. Then every ball around $x$ intersects $A$, in particular $B(x,frac12)=x$ must intersect $A$, which means $x in A$.



              So for all $A subseteq X$, $overlineA subseteq A (subseteq overlineA)$ so $A = overlineA$ and every subset $A$ is closed.






              share|cite|improve this answer









              $endgroup$

















                2












                $begingroup$

                Another way to reason: suppose $x in overlineA$ (the closure of $A$) for some arbitary subset of $A$. Then every ball around $x$ intersects $A$, in particular $B(x,frac12)=x$ must intersect $A$, which means $x in A$.



                So for all $A subseteq X$, $overlineA subseteq A (subseteq overlineA)$ so $A = overlineA$ and every subset $A$ is closed.






                share|cite|improve this answer









                $endgroup$















                  2












                  2








                  2





                  $begingroup$

                  Another way to reason: suppose $x in overlineA$ (the closure of $A$) for some arbitary subset of $A$. Then every ball around $x$ intersects $A$, in particular $B(x,frac12)=x$ must intersect $A$, which means $x in A$.



                  So for all $A subseteq X$, $overlineA subseteq A (subseteq overlineA)$ so $A = overlineA$ and every subset $A$ is closed.






                  share|cite|improve this answer









                  $endgroup$



                  Another way to reason: suppose $x in overlineA$ (the closure of $A$) for some arbitary subset of $A$. Then every ball around $x$ intersects $A$, in particular $B(x,frac12)=x$ must intersect $A$, which means $x in A$.



                  So for all $A subseteq X$, $overlineA subseteq A (subseteq overlineA)$ so $A = overlineA$ and every subset $A$ is closed.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 16 hours ago









                  Henno BrandsmaHenno Brandsma

                  116k349127




                  116k349127



























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