📂Set Theory

The Net, Moore-Smith Sequence

Definition¹

Given 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 denote it by $x_{a} = f(a) \in X$, the net $f$ is represented as $(x_{a})_{a \in A}$ or $x_{\centerdot}$². In other words,

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

Explanation

A net is also known as a Moore-Smith sequence.

A net is a generalized concept of a sequence. A sequence is defined as a function where the domain is the set of natural numbers $f : \mathbb{N} \to X$. Now, consider a scenario where instead of natural numbers, we think of the domain as a totally ordered set. We can then perceive a function with a totally ordered set as a sequence, and a function with a directed set as a net.