Inmathematics, twonon-emptysubsetsA andB of a givenmetric space (X, d) are said to bepositively separated if theinfimum

${\displaystyle \underset{a\in A,b\in B}{inf}d(a,b)>0.}$

(Some authors also specify thatA andB should bedisjoint sets; however, this adds nothing to the definition, since ifA andB have some common pointp, thend(p, p) = 0, and so the infimum above is clearly 0 in that case.)

For example, on the real line with the usual distance, theopen intervals (0, 2) and (3, 4) are positively separated, while (3, 4) and (4, 5) are not. In two dimensions, the graph ofy = 1/x forx > 0 and thex-axis are not positively separated.

• Rogers, C. A. (1998).Hausdorff measures. Cambridge Mathematical Library (Third ed.). Cambridge: Cambridge University Press. pp. xxx+195.ISBN 0-521-62491-6.

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