= Hausdorff dimension of a connected set
{c}
A <connected> subset $C$ of <Euclidean space> containing distinct points $a,b$ has <Hausdorff dimension> at least one. The <Lipschitz> map $x\mapsto|x-a|$ maps $C$ onto a set containing $[0,|b-a|]$, since <continuous images of connected spaces> are <connected>. A <Lipschitz map> cannot increase <Hausdorff dimension>, whereas an interval has <Hausdorff dimension> one. Consequently any set of <Hausdorff dimension> less than one is <totally disconnected>.
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