Are sets of rational sequences open, or closed in $\mathbb{Q}^{\omega}$?

Question

Are sets of rational sequences open, or closed in $\mathbb{Q}^{\omega}$?

I think that any subset of $\mathbb{Q}^{\omega}$ is both open and closed in $\mathbb{Q}^{\omega}$, since:

• Any subset is closed (it contains all its limit points)
• A complement of any subset is also closed

So are the subsets all clopen?

Disclaimer: $\mathbb{Q}^{\omega}$ is the space of all rational sequences, endowed with product topology.

Answer 1

No, that is not true. For example $\{(0,0,..,)\}$ is not open. If it were open then there would be a finite number of intervals $I_1,I_2,...,I_n$ around $0$ such that $x_1 \in I_1,x_2 \in I_2,..,x_n \in I_n$ implies $(x_1,x_2,..) \in \{(0,0,...)\}$ which is obviousy false.