Definition

1. Noun:
   • An interval that does not include its endpoints: In mathematics, specifically in set theory and calculus, an "unbounded interval" is a set of real numbers that extends infinitely in at least one direction and does not contain its endpoint(s). It is also commonly known as an open interval when both endpoints are finite and excluded, but "unbounded" specifically indicates the infinite extension.

Usage Examples

• Noun:
  • The set of all real numbers greater than 5, written as (5, ∞), is an unbounded interval.
  • In calculus, we often integrate over unbounded intervals, such as from 1 to infinity.

Advanced Usage

• In Real Analysis: The concept is fundamental for defining limits, integrals, and continuity over infinite domains.
  • The improper integral is defined as the limit of integrals over bounded intervals approaching an unbounded interval.
• Topological Property: An unbounded interval is an open set in the standard topology of the real number line.
  • The unbounded interval (-∞, a) is an open set in ℝ.

Variants and Related Words

• Open Interval (a, b): A bounded interval that does not include its endpoints and . This is a specific type of interval, distinct from an unbounded one which involves infinity.
• Half-open Interval: An interval that includes only one of its two endpoints (e.g., [a, b) or (a, b]). These can be bounded or unbounded if one endpoint is infinite.
• Bounded Interval: An interval that has both endpoints as finite real numbers (e.g., [a, b], (a, b], [a, b)).

Synonyms

• Infinite Interval: A direct synonym emphasizing the interval's infinite length.
• Open Ray: A term sometimes used for an unbounded interval in one direction, like (a, ∞) or (-∞, b).

Related Phrases

• Extend to Infinity: Describes the property of an unbounded interval.
  • The domain of the function is an unbounded interval extending to positive infinity.

Noun

1. an interval that does not include its endpoints