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The title says it all, really. Reading through Reed & Simon's book on functional analysis, I have now reached the chapter on topological spaces, and the notion of a net is introduced there to handle things that 'sequences' can't quite manage. It's just not clear to me why it should be called 'net'. I'm interested in this since it may help me develop my intuition for the concept.

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Why is a "field" called a "field"? Why are "germs" called "germs"? And why are "Random reals" called "Random reals"? All this, and more, in the next round of "Why do mathematical objects have their names?" (That was my favorite game show as a child!) –  Asaf Karagila Sep 1 at 21:25
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@AsafKaragila I don't mind you poking fun at my question, but I still think it's a valid and useful one to ask in many occasions. –  Danu Sep 1 at 21:26
    
I am just trying to point out that mathematical objects may or may not have names which make sense. Things like squares and diamonds, or a mouse and premouse, in set theory. All these make some, to little to no sense. And surely in other fields of mathematics this phenomenon occurs too. And besides what's "Asaf"? Or "Danu"? Or "Butch" (Well, he's American, their names don't mean ...) (Alright, that last one is a bold out reference from Pulp Fiction. But Zed's dead, baby, Zed's dead.) –  Asaf Karagila Sep 1 at 21:28
    
@AsafKaragila Zed's dead, indeed, but as you can see below, the question certainly isn't :) As a physicist, I find that most terminology makes some sense to me when explained by someone who actually understands the bigger picture involved :) Perhaps the extrapolation is unwarranted, however! –  Danu Sep 1 at 21:30
    
Let me assure you, Danu, that sometimes, there's no bigger picture. Sometimes it's just a fleeting joke, that somehow got in to the wrong place at the wrong time. –  Asaf Karagila Sep 1 at 21:31

2 Answers 2

up vote 8 down vote accepted

Imagine an actual net, like a fishing net, lying on the floor. Grab it by one vertex, and lift until the net is no longer touching the floor. The vertices of the net represent a partially ordered set. In this example, the maximum element of the partial ordering is, of course, the vertex that your fingers are holding. Now delete the vertex that your fingers are holding. The idea is that what's left is a cofinal partial ordering.

A net is just a function defined on a cofinal partially ordered set. Most often that set is the natural numbers, but not always.

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This sounds very good! I'm just wondering why we have to delete the maximum element... Which of the defining properties does it violate? –  Danu Sep 1 at 20:59
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The missing element is the $\infty$ in the symbol $\lim_{i \to \infty}$. The idea is that a sequence or net is not defined at $\infty$, but if the sequence or net converges then you can extend it to have value at $\infty$ equal to the limit. –  Lee Mosher Sep 1 at 21:26
    
It could be defined at infinity; it's ok in the definition. It only trivialises it: the limit equals the value at the maximum –  Henno Brandsma Sep 2 at 7:44

Nets and filters in real life are essentially the same things: a way to catch things. Likewise in topology, they both "catch" points of convergence. From a net, one easily defines a filter (the filter of sets the given net is "eventually in"). From a filter with the help of the axiom of choice, one finds a net by choosing elements from each set in the filter, and letting the filter itself serve as the set of indices. Convergence of a filter is the same as convergence of any of its related nets, and vice versa. So a net is rather like a filter for which we have decided where to put the "knots".

If you accept the idea that "filter" is a good name for that concept, "net" makes sense, too.

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