Properties of Intervals
The intervals are precisely the connected subsets of . It follows that the image of an interval by any continuous function is also an interval. This is one formulation of the intermediate value theorem.
The intervals are also the convex subsets of . The interval enclosure of a subset is also the convex hull of .
The intersection of any collection of intervals is always an interval. The union of two intervals is an interval if and only if they have a non-empty intersection or an open end-point of one interval is a closed end-point of the other (e.g., ).
If is viewed as a metric space, its open balls are the open bounded sets (c + r, c − r), and its closed balls are the closed bounded sets .
Any element x of an interval I defines a partition of I into three disjoint intervals I1, I2, I3: respectively, the elements of I that are less than x, the singleton, and the elements that are greater than x. The parts I1 and I3 are both non-empty (and have non-empty interiors) if and only if x is in the interior of I. This is an interval version of the trichotomy principle.
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