Zero-dimensional space
= Zero-dimensional space
A space is zero-dimensional when its topology has a base of <clopen sets>. In the Hausdorff setting these sets separate distinct points, so the space is <totally disconnected>. Conversely a compact Hausdorff <totally disconnected space> has a clopen base, by the <Šura-Bura component lemma> and finite subcovers of the complement of a neighborhood.