An inclusion-maximal separated set is one to which no further point of the ambient metric space can be added while keeping the same separation. It need not have the largest possible cardinality. In a finite space it can be obtained by repeatedly adding an admissible point.
If an inclusion-maximal separated set has separation at least , every point of the ambient metric space is at distance strictly less than from a selected point. Otherwise that point could be added. Thus upper bounds on the sizes or measures of radius- balls give lower bounds on the size of the separated set.
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