I need to prove that in a discrete metric space, every subset is both open and closed. Now, I have problem imagine how this space looks like. I think it contains of all sequences containing ones and zeros.
Now in order to prove that every subset is open, my books says that
$A \subset X $
$A$ is open if $\,\forall x \in A,\,\exists\, \epsilon > 0$ then $B_\epsilon(x) \subset A$
I was thinking that since A also will contain only zeros and ones, it must be open. Could someone help me ironing out the details? =)