What is a discrete set exactly?












2












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I'd appreciate the comments/answers including some examples in this regard. In addition, one may also ask that any infinite discrete set is a countable set?






discrete-mathematics







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  • 2




    $begingroup$
    The first is set-theoretical and the second is topological.
    $endgroup$
    – Jacky Chong
    Oct 6 '16 at 6:00






  • 1




    $begingroup$
    Every point is open.
    $endgroup$
    – Jacky Chong
    Oct 6 '16 at 6:20






  • 1




    $begingroup$
    Here's a link en.wikipedia.org/wiki/Discrete_space
    $endgroup$
    – Jacky Chong
    Oct 6 '16 at 6:20
















2












$begingroup$


I'd appreciate the comments/answers including some examples in this regard. In addition, one may also ask that any infinite discrete set is a countable set?






discrete-mathematics







share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    The first is set-theoretical and the second is topological.
    $endgroup$
    – Jacky Chong
    Oct 6 '16 at 6:00






  • 1




    $begingroup$
    Every point is open.
    $endgroup$
    – Jacky Chong
    Oct 6 '16 at 6:20






  • 1




    $begingroup$
    Here's a link en.wikipedia.org/wiki/Discrete_space
    $endgroup$
    – Jacky Chong
    Oct 6 '16 at 6:20














2












2








2


1



$begingroup$


I'd appreciate the comments/answers including some examples in this regard. In addition, one may also ask that any infinite discrete set is a countable set?






discrete-mathematics







share|cite|improve this question











$endgroup$




I'd appreciate the comments/answers including some examples in this regard. In addition, one may also ask that any infinite discrete set is a countable set?






discrete-mathematics




discrete-mathematics






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Oct 7 '16 at 11:31

























asked Oct 6 '16 at 5:59







user370634















  • 2




    $begingroup$
    The first is set-theoretical and the second is topological.
    $endgroup$
    – Jacky Chong
    Oct 6 '16 at 6:00






  • 1




    $begingroup$
    Every point is open.
    $endgroup$
    – Jacky Chong
    Oct 6 '16 at 6:20






  • 1




    $begingroup$
    Here's a link en.wikipedia.org/wiki/Discrete_space
    $endgroup$
    – Jacky Chong
    Oct 6 '16 at 6:20














  • 2




    $begingroup$
    The first is set-theoretical and the second is topological.
    $endgroup$
    – Jacky Chong
    Oct 6 '16 at 6:00






  • 1




    $begingroup$
    Every point is open.
    $endgroup$
    – Jacky Chong
    Oct 6 '16 at 6:20






  • 1




    $begingroup$
    Here's a link en.wikipedia.org/wiki/Discrete_space
    $endgroup$
    – Jacky Chong
    Oct 6 '16 at 6:20








2




2




$begingroup$
The first is set-theoretical and the second is topological.
$endgroup$
– Jacky Chong
Oct 6 '16 at 6:00




$begingroup$
The first is set-theoretical and the second is topological.
$endgroup$
– Jacky Chong
Oct 6 '16 at 6:00




1




1




$begingroup$
Every point is open.
$endgroup$
– Jacky Chong
Oct 6 '16 at 6:20




$begingroup$
Every point is open.
$endgroup$
– Jacky Chong
Oct 6 '16 at 6:20




1




1




$begingroup$
Here's a link en.wikipedia.org/wiki/Discrete_space
$endgroup$
– Jacky Chong
Oct 6 '16 at 6:20




$begingroup$
Here's a link en.wikipedia.org/wiki/Discrete_space
$endgroup$
– Jacky Chong
Oct 6 '16 at 6:20










2 Answers
2






active

oldest

votes


















2












$begingroup$

A topology has nothing to do with the set or the elements. Any set whether it is the set {1,2,3} which is finite, or the set of real functions which is a higher uncountability than the reals, can be made into a topology that can be discrete or not. A topology is how we define what subsets are open. It has nothing to do with cardinality.



A discrete space is one in which the single point sets are defined to be open.



If, for S={1,2,3} we declare: the empty set, {1}{2}{3}{1,2}{1,3}{2,3}{1,2,3} to all be open sets then that is a discrete space.



If one the other hand we declare: the empty set and {1,2,3} are open but all the other subsets are not, the is not a discrete space.



Even though the have the same elements. the elements have nothing to do with which sets are open.



Likewise, if S=$mathbb R $ and we declare the sets {x} are all open then that is a discrete space. If we declare they are not open then that is not.



You might be saying "wait a minute. We can just declare whether a set is open or not". To which the answer is... yes, we can. To a degree. (There are some restrictions.) That's what a topology is. It's a definition of openness combined with a set of elements. The actual set of elements has no say in what rules for openness we declare.






share|cite|improve this answer











$endgroup$





















    5












    $begingroup$

    The two concepts have nothing to do with each other.



    A metric space (more generally a topological space) is discrete if each point is isolated. For example, take the set of all real numbers (which, as you probably know, is uncountable) and define a new distance function
    $$operatorname d(x,y)=begin{cases}
    1text{ if }xne y,\
    0text{ if }x=y.
    end{cases}$$
    This is an uncountable discrete space.



    A space is countable if its points can be put in one-to-one correspondence with the natural numbers. For example, take the set of all rational numbers with the usual metric. It is countable but not discrete.






    share|cite|improve this answer











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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      2












      $begingroup$

      A topology has nothing to do with the set or the elements. Any set whether it is the set {1,2,3} which is finite, or the set of real functions which is a higher uncountability than the reals, can be made into a topology that can be discrete or not. A topology is how we define what subsets are open. It has nothing to do with cardinality.



      A discrete space is one in which the single point sets are defined to be open.



      If, for S={1,2,3} we declare: the empty set, {1}{2}{3}{1,2}{1,3}{2,3}{1,2,3} to all be open sets then that is a discrete space.



      If one the other hand we declare: the empty set and {1,2,3} are open but all the other subsets are not, the is not a discrete space.



      Even though the have the same elements. the elements have nothing to do with which sets are open.



      Likewise, if S=$mathbb R $ and we declare the sets {x} are all open then that is a discrete space. If we declare they are not open then that is not.



      You might be saying "wait a minute. We can just declare whether a set is open or not". To which the answer is... yes, we can. To a degree. (There are some restrictions.) That's what a topology is. It's a definition of openness combined with a set of elements. The actual set of elements has no say in what rules for openness we declare.






      share|cite|improve this answer











      $endgroup$


















        2












        $begingroup$

        A topology has nothing to do with the set or the elements. Any set whether it is the set {1,2,3} which is finite, or the set of real functions which is a higher uncountability than the reals, can be made into a topology that can be discrete or not. A topology is how we define what subsets are open. It has nothing to do with cardinality.



        A discrete space is one in which the single point sets are defined to be open.



        If, for S={1,2,3} we declare: the empty set, {1}{2}{3}{1,2}{1,3}{2,3}{1,2,3} to all be open sets then that is a discrete space.



        If one the other hand we declare: the empty set and {1,2,3} are open but all the other subsets are not, the is not a discrete space.



        Even though the have the same elements. the elements have nothing to do with which sets are open.



        Likewise, if S=$mathbb R $ and we declare the sets {x} are all open then that is a discrete space. If we declare they are not open then that is not.



        You might be saying "wait a minute. We can just declare whether a set is open or not". To which the answer is... yes, we can. To a degree. (There are some restrictions.) That's what a topology is. It's a definition of openness combined with a set of elements. The actual set of elements has no say in what rules for openness we declare.






        share|cite|improve this answer











        $endgroup$
















          2












          2








          2





          $begingroup$

          A topology has nothing to do with the set or the elements. Any set whether it is the set {1,2,3} which is finite, or the set of real functions which is a higher uncountability than the reals, can be made into a topology that can be discrete or not. A topology is how we define what subsets are open. It has nothing to do with cardinality.



          A discrete space is one in which the single point sets are defined to be open.



          If, for S={1,2,3} we declare: the empty set, {1}{2}{3}{1,2}{1,3}{2,3}{1,2,3} to all be open sets then that is a discrete space.



          If one the other hand we declare: the empty set and {1,2,3} are open but all the other subsets are not, the is not a discrete space.



          Even though the have the same elements. the elements have nothing to do with which sets are open.



          Likewise, if S=$mathbb R $ and we declare the sets {x} are all open then that is a discrete space. If we declare they are not open then that is not.



          You might be saying "wait a minute. We can just declare whether a set is open or not". To which the answer is... yes, we can. To a degree. (There are some restrictions.) That's what a topology is. It's a definition of openness combined with a set of elements. The actual set of elements has no say in what rules for openness we declare.






          share|cite|improve this answer











          $endgroup$



          A topology has nothing to do with the set or the elements. Any set whether it is the set {1,2,3} which is finite, or the set of real functions which is a higher uncountability than the reals, can be made into a topology that can be discrete or not. A topology is how we define what subsets are open. It has nothing to do with cardinality.



          A discrete space is one in which the single point sets are defined to be open.



          If, for S={1,2,3} we declare: the empty set, {1}{2}{3}{1,2}{1,3}{2,3}{1,2,3} to all be open sets then that is a discrete space.



          If one the other hand we declare: the empty set and {1,2,3} are open but all the other subsets are not, the is not a discrete space.



          Even though the have the same elements. the elements have nothing to do with which sets are open.



          Likewise, if S=$mathbb R $ and we declare the sets {x} are all open then that is a discrete space. If we declare they are not open then that is not.



          You might be saying "wait a minute. We can just declare whether a set is open or not". To which the answer is... yes, we can. To a degree. (There are some restrictions.) That's what a topology is. It's a definition of openness combined with a set of elements. The actual set of elements has no say in what rules for openness we declare.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 17 '18 at 9:01









          bof

          51.1k457120




          51.1k457120










          answered Oct 6 '16 at 7:22









          fleabloodfleablood

          69.2k22685




          69.2k22685























              5












              $begingroup$

              The two concepts have nothing to do with each other.



              A metric space (more generally a topological space) is discrete if each point is isolated. For example, take the set of all real numbers (which, as you probably know, is uncountable) and define a new distance function
              $$operatorname d(x,y)=begin{cases}
              1text{ if }xne y,\
              0text{ if }x=y.
              end{cases}$$
              This is an uncountable discrete space.



              A space is countable if its points can be put in one-to-one correspondence with the natural numbers. For example, take the set of all rational numbers with the usual metric. It is countable but not discrete.






              share|cite|improve this answer











              $endgroup$


















                5












                $begingroup$

                The two concepts have nothing to do with each other.



                A metric space (more generally a topological space) is discrete if each point is isolated. For example, take the set of all real numbers (which, as you probably know, is uncountable) and define a new distance function
                $$operatorname d(x,y)=begin{cases}
                1text{ if }xne y,\
                0text{ if }x=y.
                end{cases}$$
                This is an uncountable discrete space.



                A space is countable if its points can be put in one-to-one correspondence with the natural numbers. For example, take the set of all rational numbers with the usual metric. It is countable but not discrete.






                share|cite|improve this answer











                $endgroup$
















                  5












                  5








                  5





                  $begingroup$

                  The two concepts have nothing to do with each other.



                  A metric space (more generally a topological space) is discrete if each point is isolated. For example, take the set of all real numbers (which, as you probably know, is uncountable) and define a new distance function
                  $$operatorname d(x,y)=begin{cases}
                  1text{ if }xne y,\
                  0text{ if }x=y.
                  end{cases}$$
                  This is an uncountable discrete space.



                  A space is countable if its points can be put in one-to-one correspondence with the natural numbers. For example, take the set of all rational numbers with the usual metric. It is countable but not discrete.






                  share|cite|improve this answer











                  $endgroup$



                  The two concepts have nothing to do with each other.



                  A metric space (more generally a topological space) is discrete if each point is isolated. For example, take the set of all real numbers (which, as you probably know, is uncountable) and define a new distance function
                  $$operatorname d(x,y)=begin{cases}
                  1text{ if }xne y,\
                  0text{ if }x=y.
                  end{cases}$$
                  This is an uncountable discrete space.



                  A space is countable if its points can be put in one-to-one correspondence with the natural numbers. For example, take the set of all rational numbers with the usual metric. It is countable but not discrete.







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Oct 6 '16 at 6:36

























                  answered Oct 6 '16 at 6:21









                  bofbof

                  51.1k457120




                  51.1k457120






























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