Rational Numbers are Totally Disconnected
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Theorem
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Then $\struct {\Q, \tau_d}$ is a totally disconnected space.
Proof 1
Let $x, y \in \Q$ such that $x \ne y$.
From Between two Rational Numbers exists Irrational Number, there exists $\alpha \in \R \setminus \Q$ such that $x < \alpha < y$.
From Rational Numbers are not Connected, it follows that $x$ and $y$ belong to different components of $\Q$.
As $x$ and $y$ are arbitrary, it follows that no rational number is in the same component as any other rational number.
That is, the components of $\Q$ are singetons.
Hence the result, by definition of a totally disconnected space.
$\blacksquare$
Proof 2
Follows from:
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $6.5$: Components: Examples $6.5.1$