Properties of Spaces With Covering 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 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 where 2={0,1} is given the discrete topology. Such a space is sometimes called a Cantor cube. If I is countably infinite, is the Cantor space.
Read more about this topic: Zero-dimensional Space
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