The Unapologetic Mathematician

Mathematics for the interested outsider

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 {D} is a preorder so that for any two elements a\in D and b\in D we have some c\in D with c\geq a and c\geq b. That is, we can always find some point that’s above the two points we started with. c doesn’t have to be distinct from them, though — if a\geq b then a 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 f:A\rightarrow B and g:A\rightarrow B there is at least one arrow h:B\rightarrow C so that h\circ f=h\circ g — 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 \mathcal{J}.

Anyhow, let’s say we have such a directed set {D} at hand. If it helps, just think of \mathbb{N} with the usual order. A net in a set X is just a function \Phi:D\rightarrow X. Now we have a bunch of definitions to talk about how the image of such a function behaves. Given a subset A\subseteq X, we say that the net \Phi is “frequently” in A if for any a\in D there is a b\geq a with \Phi(b)\in A. We say that the net is “eventually” in A if there is an a\in D so that \Phi(b)\in A for all b\geq a. 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 A. The second says that we will not only land in A, but stay within A from that point on.

Next let’s equip X with a topology defined by a neighborhood system. We say that a net \Phi:D\rightarrow X converges to a point x\in X if for every neighborhood U\in\mathcal{N}(x), the net is eventually in U. In this case we call x the limit of \Phi. Notice that if {D} has a top element \omega so that \omega\geq a for all a\in D then the limit of \Phi is just \Phi(\omega). In a sense, then, the process of taking a limit is an attempt to say, “if {D} 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 x\in X is an “accumulation point” of the net \Phi if for every neighborhood U\in\mathcal{N}(x) the net is frequently in U. For instance, a sequence that jumps back and forth between two points — \Phi(n)=x for even n and \Phi(n)=y for odd n — has both x and y as accumulation points. We see in this example that if we just picked out the even elements of \mathbb{N} we’d have a convergent sequence, so let’s formalize this concept of picking out just some elements of {D}.

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 f:D'\rightarrow D between directed sets to be one so that for all d\in D there is some d'\in D' so that a'\geq d' implies that f(a)\geq d. That is, no matter “how far up” we want to get in {D}, we can find some point in D' so that the image of everything above it lands above where we’re looking in {D}. 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 \Phi:D\rightarrow X and a final function f:D'\rightarrow D we define the subnet \Phi\circ f:D'\rightarrow X.

Now the connection between accumulation points and limits is this: if x is an accumulation point of \Phi then there is some subnet of \Phi which converges to x. To show this we need to come up with a directed set D' and a final function f:D'\rightarrow D so that \Phi\circ f is eventually in any neighborhood of x. We’ll let the points of D' be pairs (a,U) where a\in D, U\in\mathcal{N}(x), and \Phi(a)\in U. We order these by saying that (a,U)\geq(b,V) if a\geq b and U\subseteq V.

Given (a,U) and (b,V) in D', then U\cap V is again a neighborhood of x, and so \Phi is frequently in U\cap V. Thus there is a c with c\geq a, c\geq b, and \Phi(c)\in U\cap V. Thus (c,U\cap V) is in D', and is above both (a,U) and (b,V), which shows that D' is directed. We can easily check that the function f:D'\rightarrow D defined by f(a,U)=a is final, and thus defines a subnet \Phi\circ f of \Phi. Now if N\in\mathcal{N}(x) is any neighborhood of x, then there is some \Phi(b)\in N. If (a,U)\geq(b,N) then \Phi(f(a,U))=\Phi(a)\in U\subseteq N. Thus \Phi\circ f is eventually in N.

Conversely, if \Phi has a limit x then x is clearly an accumulation point of \Phi.

November 19, 2007 Posted by | Point-Set Topology, Topology | 13 Comments

   

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