Definition:Net (Set Theory)

Definition

Let $\struct {S, \preceq}$ be a directed set.

Let $T$ be a set.

Let $f: S \to T$ be a mapping.

Then $f$ is a net in $T$.

Also known as

A net has historically been known as a Moore-Smith sequence, though this term is somewhat rare in modern literature.

Also see

• Definition:Net (Preordered Set)
• Definition:Sequence

Remark

In most cases, $T$ will be the underlying set of a topological space.

Source of Name

This entry was named for Eliakim Hastings Moore and Herman Lyle Smith.

Sources

• 1922: E.H. Moore and H.L. Smith: A General Theory of Limits (Amer. J. Math. Vol. 44, no. 2: pp. 102 – 121)