Nets

Why

Definition

A sequence is a function on the natural numbers; this set has two important properties: (a) we can order the natural numbers and (b) we can always go “further out.”

To elaborate on property (b): if handed two natural numbers $m$ and $n$, we can always find another, for example $\max\set{m,n}+1$, larger than $m$ and $n$. We might think of larger as “further out” from the first natural number: 1.

Now combine these two observations. A directed set is a set $D$ with a partial order $\preceq$ satisfying one additional property: for all $a, b \in D$, there exists $c \in D$ such that $a \preceq c$ and $b \preceq c$.

A net is a function on a directed set.

Examples

A sequence, then, is a net. The directed set is the set of natural numbers and the partial order is $m \preceq n$ if $m \leq n$.

Consider $\N \times \N $, and write $(a,b) \preceq (c, d)$ if $a \leq c$ and $b \leq d$. Clearly, $(\N ^2, \preceq)$ so defined is a partially ordered set. Notice that given $a = (a_1, a_2)$ and $b = (b_1, b_2)$ the point $(\max(a_1, b_1), \max(a_2, b_2))$ succeeds or is equal to both $a$ and $b$. Thus $(\N ^2, \preceq)$ is a directed set on which we can define a net.

Notation

Directed sets involve a set and a partial order. We commonly assume the partial order, and just denote the set. We use the letter $D$ as a mnemonic for directed.

For nets, we use function notation and generalize sequence notation. We denote the net $x: D \to A$ by $\{a_{\alpha }\}$, emulating notation for sequences. The use of $\alpha $ rather than $n$ reminds us that $D$ need not be the set of natural numbers.