nLab
bounded set

Bounded sets

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 EE be a metric space. A subset BEB \subseteq E is bounded if there is some real number rr such that d(x,y)<rd(x,y) \lt r for all x,yBx, 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 EE be a gauge space. A subset BEB \subseteq E is bounded if there is some real number rr such that d(x,y)<rd(x,y) \lt r for all x,yBx, y \in B and all gauging distances dd.

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 EE be a LCTVS. A subset BEB \subseteq E is bounded if whenever UEU \subseteq E is a neighbourhood of 00 then there is some real number rr such that BrUB \subseteq r U.

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

Last revised on July 5, 2017 at 01:45:40. See the history of this page for a list of all contributions to it.