= Totally separated space
A <topological space> is totally separated when any two distinct <points> are separated by a <clopen set>. Such a space has no nontrivial <connected subset>. A <zero-dimensional space> with its usual <clopen>-base definition and $T_1$ is totally separated. Conversely these conditions agree for <compact Hausdorff spaces> by the <Šura-Bura lemma> and <compactness>, although the definitions should not be conflated for arbitrary spaces. Any closed totally separated subset of the unit square is <nowhere dense>, since every nonempty relatively open subset of the square contains a nontrivial <connected> <line segment>. Restricting a <clopen set> separating the segment's endpoints would disconnect that segment.
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