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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.

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    The usual term for what you’re calling an upward filtering set is *directed set*. Nets are simply generalized sequences: $\Bbb N$ with the usual order is a (very simple) directed set, and a sequence in $X$ is a map from $\Bbb N$ into $X$. Is that enough, or are you really asking why this is an appropriate/useful generalization of sequences? – 2017-01-06
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    @BrianM.Scott Thanks, I would love to see why this is an appropriate/useful generalisation of sequences. – 2017-01-06
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    If Pedersen does not go into the motivations/uses of nets, you can find it elsewhere. For example, there is a chapter on it in Kelley's *General Topology*. – 2017-01-06
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    @Elliot: There isn’t a really short answer to that question; the notes to which [this answer](http://math.stackexchange.com/a/1636607/12042) links are one of the best introductions that I’ve seen. – 2017-01-06
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    The set of neighbourhoods of a point in reverse inclusion order is a natural directed set in a space (as the intersection of two neighbourhoods is an "upper bound" of them), and we can express all relevant notions, like closure and continuity in terms of neighbourhoods. It's also a natural filter as well, hence the theory of filters in spaces. See the notes that Brian refered to. – 2017-01-07

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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.
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    Excellent explanation, thanks! – 2017-04-11
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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 $a(n)$.