Located sets are sets from
which the distance of any point may be measured; they are used extensively in
modern constructive analysis. Here a general method is given for the construction of
all located sets on the line. It is based on a characterization of a located set in
terms of the resolution of its metric complement into a union of disjoint open
intervals. The characterization depends on a strong countability condition for
the intervals, called the locating condition. Included as a special case is the
characterization and construction of compact sets. The techniques used are in
accord with the principles of Bishop’s Foundations of Constructive Analysis,
1967.
