Let $X=\{(p_1,p_2,\dots, p_n):p_i\in \mathbb Q\}$. Is $X $ connected or disconnected?

My attempt:$X$ is connected iff any two points of $X$ are contained in a connected subset of $X$. This attempt is Not working as it is difficult to find connected subsets of $X$. Would like to prove in this approach. Topology is usual topology of $\mathbb R^n$.

  • 1
    $\begingroup$ This question doesn't make sense unless we know what topology you're talking about. Are we considering $\mathbb Q$, $\mathbb R$, $\mathbb Q_p$...? $\endgroup$ – Joshua Mundinger Nov 7 '14 at 13:50
  • $\begingroup$ In particular, it's connected in $X$ under the indiscrete topology. $\endgroup$ – Joshua Mundinger Nov 7 '14 at 13:51

Hint: consider the hyperplanes $x_i=k$, ($i$ fixed, $k\in\Bbb R\setminus\Bbb Q$).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.