Intuition behind a net

Question

In Analysis Now, Pedersen defines a net to be a pair $(\Lambda, i)$, where $\Lambda$ is an upward filtering ordered set and $i$ is a map from $\Lambda$ into $X$.

I don't understand the intuition for this however. I'm aware that an upward filtering set is a set $X$ such that for every pair in $X$ there is an upper bound in that pair.

Answer 1

A net is simply a sequence where we have relaxed what the indexing set is. Recall that sequences are essentially functions from the naturals $\mathbb{N}$ to some space $X$. If we change $\mathbb{N}$ to be some other set,then the structure that arises is referred to as a net.

What is the point of this abstraction? Well, you should know that for metric spaces, we have two equivalent definitions of compactness, sequential compactness and topological compactness. That is,

• Sequential compactness - Every bounded sequence has a convergent subsequence.
• Topological Compactness - Every open cover has a finite sub cover.

When we leave the world of metric spaces however, the definitions are not equivalent. We do still have the following equivalent formulations of compactness,

• Every bounded net has a convergent subnet
• Every open cover has a finite sub cover.

Answer 2

Maybe this helps: a sequence $\{a_n\}$ on $X$ is a pair $(\mathbb N, a)$, where $a:\mathbb N\to X$ is a map. We often write $a_n$ instead of $a(n)$.