Net (mathematics)

In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a function whose domain is a directed set. The codomain of this function is usually some topological space. Nets directly generalize the concept of a sequence in a metric space. Nets are primarily used in the fields of analysis and topology, where they are used to characterize many important topological properties that (in general) sequences are unable to characterize (this shortcoming of sequences motivated the study of sequential spaces and Fréchet–Urysohn spaces). Nets are in one-to-one correspondence with filters.

History

The concept of a net was first introduced by E. H. Moore and Herman L. Smith in 1922. The term "net" was coined by John L. Kelley.

The related concept of a filter was developed in 1937 by Henri Cartan.

Definitions

A directed set is a non-empty set

A

{\displaystyle A}

together with a preorder, typically automatically assumed to be denoted by

≤

{\displaystyle \,\leq \,}

(unless indicated otherwise), with the property that it is also (upward) directed, which means that for any

a

,

b

∈

A

,

{\displaystyle a,b\in A,}

there exists some

c

∈

A

{\displaystyle c\in A}

such that

a

≤

c

{\displaystyle a\leq c}

and

b

≤

c

.

{\displaystyle b\leq c.}

In words, this property means that given any two elements (of

A

{\displaystyle A}

), there is always some element that is "above" both of them (greater than or equal to each); in this way, directed sets generalize the notion of "a direction" in a mathematically rigorous way.