nLab bounded set

Idea

The term ‘bounded’ has several meaning in different branches of mathematics. For a general axiomatic approach to boundedness, see bornological set. Here we list definitions in various fields.

In metric spaces

Definition

Let $E$ be a metric space. A subset $B \subseteq E$ is bounded if there is some real number $r$ such that $d(x,y) \lt r$ for all $x, y \in B$.

This generalises immediately to pseudometric spaces, quasimetric spaces, extended metric spaces, and most generally to Lawvere metric spaces.

We can also generalise to gauge spaces:

Definition

Let $E$ be a gauge space. A subset $B \subseteq E$ is bounded if there is some real number $r$ such that $d(x,y) \lt r$ for all $x, y \in B$ and all gauging distances $d$.

This generalises immediately to quasigauge spaces.

The family of all bounded sets of a quasigauge space (and hence of the more particular kinds of spaces above) defines a bornology on its underlying set.

In topological vector spaces

Definition

Let $E$ be a LCTVS. A subset $B \subseteq E$ is bounded if whenever $U \subseteq E$ is a neighbourhood of $0$ then there is some real number $r$ such that $B \subseteq r U$.

The family of all bounded sets of a LCTVS defines a bornology on its underlying set.

Related concepts

• totally bounded metric space

• bounded function