# Is the set $\{(p_1,p_2,\dots, p_n):p_i\in \mathbb Q\}$ connected?

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 $\mathbb R^n$.

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

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