prove or disprove:

Let (X, T ) be a topological space. Then T is the discrete topology on X iff Bd(A) = ∅ for every A ⊆ X.

I think it is false statement, but I do not know how can I come with counterexample any help please


closed as off-topic by Willie Wong, Brevan Ellefsen, Magdiragdag, user91500, user223391 Apr 26 '17 at 2:21

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When unable to find a counterexample, one should always try to give proof. The statement is indeed true. If $T$ is not the discrete topology, we can take a subset $A \subseteq X$ which is not closed. Then $\overline{A}$ properly contains $A$ and so $\partial A = \overline{A}\setminus A^\circ$ is non-empty.

  • $\begingroup$ you mean the statement is true? $\endgroup$ – rian asd Apr 25 '17 at 0:42
  • $\begingroup$ so, how can I PROVE it since it has two side (iff) $\endgroup$ – rian asd Apr 25 '17 at 0:44
  • $\begingroup$ I literally said the statement is true. I proved one direction for you. You can try proving the other direction. It is easier. $\endgroup$ – mathworker21 Apr 25 '17 at 1:07
  • $\begingroup$ ok, thank you so much $\endgroup$ – rian asd Apr 25 '17 at 1:31
  • $\begingroup$ I am waiting you $\endgroup$ – rian asd Apr 25 '17 at 1:55

A useful theorem to prove is
bd A is empty iff A is clopen.
Next show every subset of a discrete space is clopen.


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