Zero-dimensional space

Properties of spaces with small inductive dimension zero

• A zero-dimensional Hausdorff space is necessarily totally disconnected, but the converse fails. However, a locally compact Hausdorff space is zero-dimensional if and only if it is totally disconnected. (See (Arhangel'skii & Tkachenko 2008, Proposition 3.1.7, p.136) for the non-trivial direction.)
• Zero-dimensional Polish spaces are a particularly convenient setting for descriptive set theory. Examples of such spaces include the Cantor space and Baire space.
• Hausdorff zero-dimensional spaces are precisely the subspaces of topological powers ${\displaystyle 2^I}$ where ${\displaystyle 2=\{0,1\}}$ is given the discrete topology. Such a space is sometimes called a Cantor cube. If I is countably infinite, ${\displaystyle 2^I}$ is the Cantor space.

Manifolds

All points of a zero-dimensional manifold are isolated.