Basic Analysis I — Real Analysis · Problem 7.2.5

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · open-and-closed-sets

Problem

Take

\mathbb{Q}

with the standard metric,

d(x,y) = |x-y|

, as our metric space. Prove that

\mathbb{Q}

is totally disconnected, that is, show that for every

x, y \in \mathbb{Q}

with

x \neq y

, there exist two open sets

U

and

V

such that

x \in U

y \in V

U \cap V = \emptyset

, and

U \cup V = \mathbb{Q}