Interval (mathematics)
Properties

The intervals are precisely the connected subsets of R. {\displaystyle \mathbb {R}.} It follows that the image of an interval by any continuous function from R {\displaystyle \mathbb {R} } to R {\displaystyle \mathbb {R} } is also an interval. This is one formulation of the intermediate value theorem.

The intervals are also the convex subsets of R. {\displaystyle \mathbb {R}.} The interval enclosure of a subset X ⊆ R {\displaystyle X\subseteq \mathbb {R} } is also the convex hull of X. {\displaystyle X.}

The closure of an interval is the union of the interval and the set of its finite endpoints, and hence is also an interval. (The latter also follows from the fact that the closure of every connected subset of a topological space is a connected subset.) In other words, we have

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, for example (a, b) ∪ = (a, c ]. {\displaystyle (a,b)\cup =(a,c].}

If R {\displaystyle \mathbb {R} } is viewed as a metric space, its open balls are the open bounded intervals (c + r, c − r), and its closed balls are the closed bounded intervals. In particular, the metric and order topologies in the real line coincide, which is the standard topology of the real line.

Any element x of an interval I defines a partition of I into three disjoint intervals I, I, I: respectively, the elements of I that are less than x, the singleton = { x }, {\displaystyle =\{x\},} and the elements that are greater than x. The parts I and I 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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