Sequences are a very useful tool but they are inadequate in some astonishingly elementary applications.
Take the case of pointwise convergence of functions (studied earlier in this blog here and here. )
Let’s look at the very simple example: for all . Yes, this is just the constant function.
Now consider a set which consists of all functions if where is some finite set of points, and otherwise. Remember there exists such that for all we have . Note: the can vary with .
Now recall that . Now what topology are we using in , since this is a product space? For pointwise convergence, we use the product topology in which the open sets are the usual open sets for a finite number of values in and the entire real line for the remaining values.
If this seems strange, consider the easier case where represents the natural numbers. Then we can view elements of as sequences, each of which takes a real value. And the open sets here will be the “sequences” where the are copies of the real line, except for a finite number of indices in which case the can be some arbitrary open set in the real line.
For , we index by the real numbers instead of by .
So, in terms of pointwise convergence, this means for every the is associated with the open set in the product topology where the real line factor associated with that value of has the usual real line open sets and the remaining factors of just have the whole real line; in short, they don’t really matter when figuring out of this function converges AT THIS VALUE OF .
So with the product topology in place, look at our for all . Given ANY open set , we see that . For example, if for , and is the open set which corresponds to in the factor and the real line elsewhere, then . So, we conclude that is in the topological closure of .
But THERE IS NO SEQUENCE IN which converges to . In fact, it is relatively easy to see that if is any sequence in and if , then is zero in at most a countable number of points. That is, if we use the Lebesgue integral, (of course, might not be Riemann integrable).
So sequences cannot reach a point in the closure. For the experts: this shows that in the product topology is not a first countable topological space; that is, its topology has no countable neighborhood basis. This also implies that it is not a metric space (or, more precisely, can’t be made into a metric space).
But I am digressing. The point is that, in situations like this, we want another tool to take the place of a sequence. That will be called a net.
Nets and Directed Sets
Roughly, a net is a “sequence like” thing that can be indexed by an uncountable set. But this indexing set needs to have a “direction like” quality to it. So, what works are “directed sets”.
A directed set is a collection of objects with a relation that satisfy the following properties:
1) (reflexive property)
2) If and then (transitive property)
3) Given and there exists where and (direction)
Note: though a directed set could be an ordered set (say, with the usual order relation) or a partially ordered set (say, subsets ordered by inclusion), they don’t have to be. For example: one can form a directed set out of the complex numbers by declaring if . Then note and but .
Now a net is a map from a directed set into a space. It is often denoted by ( is an element in the index set, which is a directed set). So, a real valued net indexed by the reals is, well, an object in .
Now given a set in a topological space, we say that a net is eventually in if there is an index such that, for all , and we say that is eventually in . We say that a net if for all open sets we have is eventually in .
Now getting back to our function example: we CAN come up with a net in that converges to our function ; we merely have to be clever at how we choose our index set though. One way: make a directed set by declaring if . Now if we take any neighborhood of in the product topology, (remember that this consists of the product of a finite number of the usual open set in the real line with an infinite number of copies of the real line), we have elements of this net eventually in this open set, namely the functions which are zero for the values of that correspond to those open sets. (see here for a couple of ways of doing this)
This demonstrates the usefulness of nets. Note that trying to use a “sequence idea” by just starting with a function that is zero at exactly one point and then going to a function that is zero at two points, three points,…can only get you to a function that is zero at a countable number of points, which is NOT in . That is, one “leaves prior to getting to where one wants to go, which is a function that is zero at all points of the real line.
On the other hand, a directed set can “start” at an uncountable number of elements of to begin with and get to being eventually in any basic open set containing in a finite number of steps. Of course, one must allow for an uncountable number of sequence like paths to get into any of the uncountable number of basic open sets, but each path consists of only a finite number of steps.