Intuition behind a net 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. 
 A: 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 $a(n)$. 
A: 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, 


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*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, 


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*Every bounded net has a convergent subnet 

*Every open cover has a finite sub cover. 

