Dawg, I am a mathematics undergraduate student looking to get deeply involved in – guess – math. So this blog will be about my experiences as a mathematician.

Also wordpress can do \LaTeX and that makes me so excited.

I don’t think I’ll be posting much cutting edge stuff here, being not a super-ultra-badass-of-math, but rather things that catch my attention as I learn about more math. Please point out my errors or discuss what is posted, as the purpose of this blog is to provide a platform for my exploration of mathematics (and yours, I should hope).

And to kick us off: today, I discovered nets in Munkres’ text. A net is kinda like a generalization of sequences in metrizable spaces for general topological spaces. So what does that mean?

Well, let’s try generalizing sequences and see how far we get. As a refresher, sequences in some set S are essentially maps \mathbb{N} \rightarrow S. A convergent sequence satisfies this: there exists an x such that for every \epsilon>0, there exists an N \in \mathbb{N} such that n \geq N implies d(x_n, x) < \epsilon (and of course x is called the limit point). That’s great and all, but this definition depends on the existence of the metric, and so metrizability in spaces is a requirement for this definition of convergence to be applicable. Let’s attempt to revise this definition in more generality: a sequence converges to x iff for every neighborhood U of x, there exists an N \in \mathbb{N} such that n \geq N implies x_n \in U. Are there any problems with this definition?

In a nutshell, yes, or else nets wouldn’t have been invented. Still, all I did was transfer the original notion of how general topology managed to avoid the notion of epsilon by using neighborhoods – intuitively, I want this definition to work. It won’t work in general, though.

Oh, and by ‘work,’ I mean that this definition should be powerful enough so that the following conditions are equivalent:

  1. f is continuous
  2. x_n \rightarrow x \Rightarrow f(x_n) \rightarrow f(x)

We can show (1)=>(2) fairly simply for all spaces, but (2)=>(1) does not hold in general. However, it does hold for first-countable spaces: if we assume that f is not continuous, we can use first-countability to construct a convergent sequence x_n contained in a countable neighborhood basis of x which when mapped onto the codomain does not converge to f(x).

So nets generalize this by widening the index set of the sequence to include uncountable sets so the first-countable requirement on the topological space is removed for (2)=>(1). To define a net, we need a bit of set theory:

A partially ordered set, or a poset, is a set with a relation \leq such that:

  1. a \leq a for every a.
  2. a \leq b and b \leq a iff a = b
  3. a \leq b and b \leq c only if a \leq c .

Note that the relation need not be defined between every pair of elements in the set to satisfy the definition. A partial order does not impose a strong enough structure for use in describing convergence (consider a partial order defined solely on reflexive pairs – wouldn’t be useful for this purpose at all). ¬†We could go ahead then and use totally ordered sets, but in fact a weaker structure would prevail:

A directed set is a poset with the property that for any two elements a, b, there exists an element c such that a \leq c and b \leq c.

So now, a net in X can be defined as a map from J to X. It really does look like a net if you care to mentally picture it, actually. And convergence is pretty much defined the same way – in fact, sequences are nets with the natural numbers representing the index set and the good ol’ less-than-or-equal-to as a total order. Proofs using nets involve an extra complication: generally you will need to prove things for nets in general, and that means stacking a bit of set theory on top of whatever topology you’re doing. On the other hand, you have a nice analog for sequences in topological spaces.

Filters are even more general things, I hear. Maybe I’ll get into that on a different day.