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Discrete topology and discrete subspaces

  1. Mar 11, 2013 #1
    1. The problem statement, all variables and given/known data
    If A is a subspace of X and A has discrete topology does X have discrete topolgy?Also if X has discrete topology then does it imply that A must have discrete topology?



    3. The attempt at a solution
    My understanding of discrete topology suggests to me that if A is discrete it doesn't imply that X is also. example A the natural numbers as a subspace of X the real numbers with the euclidean metric.

    I feel the reverse is true. For X to be discrete does every singleton in the metric space have to be open? ie: is it correct to say that the a metric topology is discrete if and only if each singleton in the metric space is open?
    I think the answer is yes in which case each singlton in X is open meaning any subspace is also discrete. Since the subspace would be a collection of singletons and unions of singletons.

    Also for a set to be open is true to say: a set A a subset of X is open if and only if for all a in A d(a,X-A)>0?
     
    Last edited: Mar 11, 2013
  2. jcsd
  3. Mar 11, 2013 #2

    Dick

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    That all seems pretty ok to me.
     
  4. Mar 12, 2013 #3
    Thanks for confirming. Very hard to be sure since there always seems to be a catch.
     
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