Is the following statement true? Completeness is a topological property

No. (0,1) and ℝ Homeomorphic, but ℝ is complete and (0,1) is not (1/n is Cauchy).

Counterexample

Is this true?

Completeness is a topological property

No — it is false.

The counterexample

(0,1) and $\mathbb{R}$

Why it fails

Homeomorphic, but $\mathbb{R}$ is complete and (0,1) is not (1/n is Cauchy).

Keep practicing

Finite-dimensional intuition

Something true in ℝⁿ, or in a nice space, assumed in general.

Watch it happen

Baire category, and the argument it powers

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.