Nets, Part I
And now we come to my favorite definition of a topology: that by nets. If you’ve got a fair-to-middling mathematical background you’ve almost certainly seen the notion of a sequence, which picks out a point of a space for each natural number. Nets generalize this to picking out more general collections of points.
The essential thing about the natural numbers for sequences is that they’re “directed”. That is, there’s an order on them. It’s a particularly simple order since it’s total — any two elements are comparable — and the underlying set is very easy to understand. We want to consider more general sorts of “directed” sets, and we define them as follows: a directed set is a preorder so that for any two elements and we have some with and . That is, we can always find some point that’s above the two points we started with. doesn’t have to be distinct from them, though — if then is just such a point.
In categorical terms this is not quite the same as saying that our preorder has coproducts, since we don’t require any sort of universality here. We might say instead that we have “weak” binary coproducts, but that might be inessentially multiplying entities, and Ockham don’t play that. However, if we also throw in the existence of “weak” coequalizers — for a pair of arrows and there is at least one arrow so that — we get something called a “filtered” category. Since there’s no such thing as a pair of distinct parallel arrows in a preorder, this adds nothing in that case. However, filtered categories show up in the theory of colimits in categories. In fact originally colimits were only defined over filtered index categories .
Anyhow, let’s say we have such a directed set at hand. If it helps, just think of with the usual order. A net in a set is just a function . Now we have a bunch of definitions to talk about how the image of such a function behaves. Given a subset , we say that the net is “frequently” in if for any there is a with . We say that the net is “eventually” in if there is an so that for all . For sequences, the first of these conditions says that no matter how far out the sequence we go we can find a point of the sequence in . The second says that we will not only land in , but stay within from that point on.
Next let’s equip with a topology defined by a neighborhood system. We say that a net converges to a point if for every neighborhood , the net is eventually in . In this case we call the limit of . Notice that if has a top element so that for all then the limit of is just . In a sense, then, the process of taking a limit is an attempt to say, “if did have a top element, where would it have to go?”
Now, a net may not have a limit. A weaker condition is to say that is an “accumulation point” of the net if for every neighborhood the net is frequently in . For instance, a sequence that jumps back and forth between two points — for even and for odd — has both and as accumulation points. We see in this example that if we just picked out the even elements of we’d have a convergent sequence, so let’s formalize this concept of picking out just some elements of .
For sequences you might be familiar with finding subsequences by just throwing out some of the indices. However for a general directed set we might not be left with a directed set after we throw away some of its points. Instead, we define a final function between directed sets to be one so that for all there is some so that implies that . That is, no matter “how far up” we want to get in , we can find some point in so that the image of everything above it lands above where we’re looking in . For sequences this just says that no matter how far out the natural numbers we march there’s still a point ahead of us that we’re going to keep. Then, given a net and a final function we define the subnet .
Now the connection between accumulation points and limits is this: if is an accumulation point of then there is some subnet of which converges to . To show this we need to come up with a directed set and a final function so that is eventually in any neighborhood of . We’ll let the points of be pairs where , , and . We order these by saying that if and .
Given and in , then is again a neighborhood of , and so is frequently in . Thus there is a with , , and . Thus is in , and is above both and , which shows that is directed. We can easily check that the function defined by is final, and thus defines a subnet of . Now if is any neighborhood of , then there is some . If then . Thus is eventually in .
Conversely, if has a limit then is clearly an accumulation point of .