A (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set

Intervals - Classification

The intervals of real numbers can be classified into the eleven different types listed below, where a and b are real numbers, and 𝑎 < 𝑏:

• Empty Interval: [𝑏,𝑎]=(𝑏,𝑎)=[𝑏,𝑎)=(𝑏,𝑎]=(𝑎,𝑎)=[𝑎,𝑎)=(𝑎,𝑎]={}=∅
• Degenerate Interval: [a,a]={𝑎} is any set consisting of a single real number
• Proper Intervals - is a real interval that is neither empty nor degenerate and has infinitely many elements
  • Bounded/Finite Intervals -
    • Open Interval: (𝑎,𝑏)={𝑥 | 𝑎<𝑥<𝑏}
    • Closed Interval: [𝑎,𝑏]={𝑥 | 𝑎≤𝑥≤𝑏}
    • Left-Closed, Right-Open Interval: [𝑎,𝑏)={𝑥 | 𝑎≤𝑥<𝑏}
    • Left-Open, Right-Closed Interval: (𝑎,𝑏]={𝑥 | 𝑎<𝑥≤𝑏}
  • Half-Bounded Intervals - intervals that are bounded at only one end
    • Left-Bounded & Right-Unbounded Intervals:
      • Left-Open Interval: (𝑎,+∞)={𝑥 | 𝑥>𝑎}
      • Left-Closed Interval: [𝑎,∞)={𝑥 | 𝑥≥𝑎}
    • Left-Unbounded & Right-Bounded Intervals:
      • Right-Open Interval: (-∞,𝑏)={𝑥 | 𝑥<𝑏}
      • Right-Closed Interval: (-∞,𝑏]={𝑥 | 𝑥≤𝑏}
  • Unbounded Interval at both ends (simultaneously open and closed): (-∞,+∞)=ℝ

Intervals - Terminology

• open interval does not include its endpoints, and is indicated with parentheses. (0,1)
• closed interval is an interval which includes all its limit points, and is denoted with square brackets. [0,1]
• half-open interval includes only one of its endpoints, and is denoted by mixing the notations for open and closed intervals. (0,1] and [0,1)
• left-bounded interval is an interval where all its elements are larger than some real number
• right-bounded interval is an interval where all its elements are smaller than some real number
• bounded/finite interval is an interval that is both left-bounded and right-bounded (empty set is bounded)
• unbounded interval is any interval that is not a bounded interval (the set of all reals is the only interval that is unbounded at both ends)
  • half-bounded interval is an interval that either left-bounded or right-bounded
• diameter (length, width, measure, range, or size) of interval is equal to the absolute difference between the endpoints:
  • diameter of a bounded interval is 𝑏-𝑎
  • diameter of unbounded intervals is +∞
  • diameter of empty interval is 0 or undefined
• radius is half the diameter
• centre (midpoint)
  • centre of a bounded interval is (𝑎 + 𝑏)/2
  • centre of unbounded intervals is undefined
  • centre of empty interval is undefined
• interior of an interval 𝐼 is the largest open interval that is contained in 𝐼; it is also the set of points in 𝐼 which are not endpoints of 𝐼
• closure of an interval 𝐼 is the smallest closed interval that contains 𝐼; which is also the set 𝐼 augmented with its finite endpoints
• interval enclosure or interval span of some set 𝑋 of real numbers, is the unique interval that contains 𝑋 and does not properly contain any other interval that also contains 𝑋
• subinterval of interval 𝐽 is some interval 𝐼 such that: 𝐼 is a subset of 𝐽
• proper subinterval of interval 𝐽 is some interval 𝐼 such that: 𝐼 is a proper subset of 𝐽

Note on Conflicting Terminology

The terms segment and interval have been employed in the literature in two essentially opposite ways, resulting in ambiguity when these terms are used. The Encyclopedia of Mathematics defines interval (without a qualifier) to exclude both endpoints (i.e., open interval) and segment to include both endpoints (i.e., closed interval), while Rudin’s Principles of Mathematical Analysis calls sets of the form [a, b] intervals and sets of the form (a, b) segments throughout. These terms tend to appear in older works; modern texts increasingly favor the term interval (qualified by open, closed, or half-open), regardless of whether endpoints are included