Irrational Number Space is Complete Metric Space

Theorem

Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$.

Then $\struct {\R \setminus \Q, \tau_d}$ is a complete metric space.

Proof

Let $d: \R \times \R \to \R$ be the Euclidean plus metric:

$\ds \map d {x, y} := \size {x - y} + \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {\size {x - r_j} } - \max_{j \mathop \le i} \frac 1 {\size {y - r_j} } } }$

Let $\sequence {x_n}$ converge in $\struct {\R \setminus \Q, \tau_d}$ to a rational number.

For each $x_i$ in $\sequence {x_n}$, there exists $j > i$ such that $\map d {x_i, x_j} \ge \size {x_i - x_j} + 2^{-k}$ for some $k$.

Hence $\sequence {x_n}$ is not a Cauchy sequence.

Thus if $\sequence {x_n}$ is a Cauchy sequence it converges to an irrational number.

This needs considerable tedious hard slog to complete it. In particular: Tighten up which metrics and which metric spaces are being converged in or not, and establish the direct relevance of the metric $d$. This has been lifted more or less verbatim from Steen and Seebach and isn't fully understood. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $31$. The Irrational Numbers: $5$