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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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.
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.