📂Set Theory

Net, Moore-Smith Sequence

Definition¹

For a set $X$ and a directed set $A$, a function $f : A \to X$ from $A$ to $X$ is called a net.

Notation

For each $a \in A$, if we write $x_{a} = f(a) \in X$, then the net $f$ is denoted by $(x_{a})_{a \in A}$ or $x_{\centerdot}$². That is,

$$ x_{\centerdot} : A \to X \\ x_{\centerdot}(a) = x_{a} = f(a) $$

Explanation

A net is also called a Moore-Smith sequence.

A net is a generalized notion of a sequence. A sequence refers to a set $f : \mathbb{N} \to X$ whose domain is the natural numbers; let us think of this domain as a totally ordered set rather than the natural numbers. Then we can regard a function whose domain is a totally ordered set as a sequence, and a function whose domain is a directed set as a net.