Editing Zero-dimensional space (section)

== 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 {{harv|Arhangel'skii|Tkachenko|2008|loc=Proposition 3.1.7, p.136}} for the non-trivial direction.)
* Zero-dimensional [[Polish space]]s are a particularly convenient setting for [[descriptive set theory]]. Examples of such spaces include the [[Cantor space]] and [[Baire space (set theory)|Baire space]].
* Hausdorff zero-dimensional spaces are precisely the [[Subspace topology|subspaces]] of topological [[power set|powers]] <math>2^I</math> where <math>2=\{0,1\}</math> is given the [[discrete topology]]. Such a space is sometimes called a [[Cantor cube]]. If {{mvar|I}} is [[countable set|countably infinite]], <math>2^I</math> is the Cantor space.