Definition

1. Noun:
   • A set of real numbers between two given numbers, excluding the endpoints themselves: An "open interval" is a fundamental concept in mathematics, specifically in calculus and real analysis. It consists of all the numbers greater than one endpoint and less than the other endpoint. The endpoints are not part of the set.

Usage

• The term is used primarily in mathematical contexts to describe a continuous range of values.
• It is denoted using parentheses, e.g., (a, b), where 'a' and 'b' are the excluded endpoints.
• It contrasts with a "closed interval," which includes its endpoints and is denoted with square brackets, e.g., [a, b].

Examples

• Noun:
  • The function is defined for all x in the open interval (0, 5). (This means x can be any number greater than 0 and less than 5, but not 0 or 5.)
  • The solution lies within the open interval (-1, 1).
  • We are considering values in the open interval between the two critical points.

Advanced Usage

• "Open interval" in topology: In more advanced mathematics, an open interval is a basic example of an "open set" in the standard topology on the real number line.
• Extended real numbers: The concept can be extended to include intervals like (a, ∞) or (-∞, b), which are also considered open intervals.

Variants and Related Words

• Interval (n): A general term for a set of numbers between two endpoints.
• Closed Interval (n): An interval that include its endpoints, denoted [a, b].
• Half-open Interval (n): An interval that includes only one of its endpoints, e.g., [a, b) or (a, b].
• Bounded Interval (n): An interval where both endpoints are finite numbers.
• Open Set (n): A more general topological concept of which an open interval is a primary example.

Synonyms

• (a, b) notation: The standard symbolic representation.
• Strict interval: A less common term emphasizing the exclusion of endpoints.

Related Phrases

• "Over an open interval": Used to describe the domain or range of a function or the validity of a statement.
  • The derivative is positive over the open interval.
• "Contained in an open interval": Describes where a particular value or set of values lies.
  • The root of the equation is contained in the open interval (2, 3).

Noun

1. an interval that does not include its endpoints