In mathematics, two

non-empty In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...

subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...

s ''A'' and ''B'' of a given

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...

(''X'', ''d'') are said to be positively separated if the

infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...

\inf_ d(a, b) > 0.

(Some authors also specify that ''A'' and ''B'' should be

disjoint sets In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A ...

; however, this adds nothing to the definition, since if ''A'' and ''B'' have some common point ''p'', then ''d''(''p'', ''p'') = 0, and so the infimum above is clearly 0 in that case.) For example, on the real line with the usual distance, the

open interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Othe ...

s (0, 2) and (3, 4) are positively separated, while (3, 4) and (4, 5) are not. In two dimensions, the graph of ''y'' = 1/''x'' for ''x'' > 0 and the ''x''-axis are not positively separated.

References

* {{cite book , author = Rogers, C. A. , title = Hausdorff measures , edition = Third , series = Cambridge Mathematical Library , publisher = Cambridge University Press , location = Cambridge , year = 1998 , pages = xxx+195 , isbn = 0-521-62491-6 Metric geometry