Nets and Continuity
Okay, so why have we been talking about nets? Because continuous functions look great in terms of nets!
First I’ll give you the answer: a function 
is continuous if and only if 
. To be a little more clear, let’s write 
for 
. Then 
. That is, a continuous function preserves the limits — and more generally the accumulation points — of all nets. Now this looks a lot more like algebra than that messy business of pulling back open sets!
We can even get a little finer and say that a function is continuous at a point 
if every net in 
that converges to 
gets sent to a net in 
converging to 
. Then we say that a function is continuous if it is continuous at all points of . This should remind us of how we defined continuity at a point by using neighborhood systems, and so we’ll show the equivalence of that definition of continuity and our new one.
So, let and have the neighborhood systems 
and 
, respectively. We’ll assume that for every neighborhood 
there is a neighborhood 
with 
. Now if we take a net 
converging to , we must show that 
is eventually in 
for all . But for each such neighborhood of we have a neighborhood , and we know that is eventually in 
. Then must be eventually in , and so converges to .
On the other hand, let’s suppose that there is some neighborhood 
of so that no neighborhood of completely fit into . We’ll construct a net converging to , but whose image doesn’t converge to . For our directed set we take the neighborhood filter 
itself, ordered by inclusion. That is, 
if 
. Then since 
there must be some point 
with 
. We pick any such point as the value of our net at . Clearly the net 
is eventually in every neighborhood of , and so the net converges to . But just as clearly, since 
is not eventually in , the image net can’t converge to .
So nets give us a very “algebraic” picture of topological spaces. A topological space is a set equipped with a (partially-defined) rule that sends every convergent net 
to its limit point in , and continuous maps are those which preserve this rule. Still, there’s something different here. Since taking the limit only works on some nets, this “preservation” is to be read in a more logical sense: if the net converges then the image net converges, and we know the answer. However, the image net could easily converge without the original net converging, and then we have no idea what its limit is. This is in contradistinction to the case for algebraic structures, where the algebraic operations are always defined and the connection between source and target structures feels a lot tighter.
There’s also a tantalizing connection to category theory, in that our directed sets are categories of a sort. Clearly I’d like to think of a net as some sort of functor, and the limit of a net as being the limit of this functor. But I don’t really see what the target category should be. I could take objects to be points of , but then what are the morphisms? And if the objects aren’t points of , what are they? How does this process of taking a limit correspond to the categorical one?
November 21, 2007 -
Posted by John Armstrong |
Point-Set Topology, Topology
The Unapologetic Mathematician 
I know I said this already, in a comment to one of your previous posts, but if you want an algebra-style description of topological spaces and continuous maps that’s simpler than the one with nets you can just use closure rewritten as a relation:
A topological space X is just a set equipped with a relation between elements and subsets (this relation, then, is a subset of X x P(X), where P(X) is the powerset of X), satisfying Kuratowski’s closure axioms [1]; and a continuous function is just set theoretic function preserving that relation.
I think that a subset of X x P(X) is an easier thing to talk about preserving than a rule assigning limits to some of the very many nets in the space.
Footnote [1]: To be completely explicit, its the induced closure operator that should satisfy the Kuratowski axiom. If you call the relation R, the closure of a subset A of X is the set of points p such that p R A.
As I recall, you said something about “closeness”, which wasn’t explicitly defined. Do you mean to say the relation is “
is in the closure of 
“? That might work, but I don’t know offhand.
Anyhow, one thing I’ll say for nets is that they not only give a criterion for continuity that involves preserving something, but they also give a very local criterion. That is, we naturally talk about continuity or discontinuity at a point. This is similar to the neighborhood definition, but seems a bit more elegant to me.
Yes, I mean “x is in the closure of A”. Sorry if wasn’t clear (twice).
That this works, i.e., that defining closeness as “x is in the closure of A” makes continuity equivalent to closeness preserving, is very easy to prove. Once you unwind the definition of “closeness preserving” what’s left to prove is the very simple statement that
f is continuous iff for every A, f(cl A) is contained in cl f(A).
Maybe I didn’t read carefully your assumptions, but you wrote ‘its limit’, or ‘the limit’ in place of ‘a limit’, in those three interesting posts (and in the definition of subnets there is an $a$ instead of $a’$).
I would consider “nets on X” as a comma category (its monomorphisms would be final functions between the index spaces). Therefore a function f from X to Y would induce a functor (the direct image) from nets on X to nets on Y. If one considers Lim_X (or equivalently Accum_X) as a functor from “nets on X” to “subsets of X”, then f being continuous means $f \circ Lim_X \subseteq Lim_Y \circ f$ (which is Omar’s point of view).
By the way, the “contradistinction to the case for algebraic structures” you mention is illustrated by the fact the a bijective morphism in an algebraic structure is an isomorphism, but a continuous bijection need not be an homeomorphism.
“There’s also a tantalizing connection to category theory, in that our directed sets are categories of a sort. Clearly I’d like to think of a net as some sort of functor, and the limit of a net as being the limit of this functor. But I don’t really see what the target category should be. I could take objects to be points of X, but then what are the morphisms? And if the objects aren’t points of X, what are they? How does this process of taking a limit correspond to the categorical one?”
Prof Peter Johnstone gave a course entitled something like “pointless toplogy” at Cambridge, but I was a year too young to know about it. I suspect the answer lies therein; certainly thinking about the points of topological spaces is not as categorical as thinking about the structure of the collection of their open sets, which forms an algebraic structure called a locale.
Therefore, I suspect that the target category is the locale of open sets, and then your “limits” are somehow directed meets in this lattice, and “directed meet” is another way of saying “filtered limit”.
When you started talking about limits of directed structures I thought you were going to go down this route, perhaps heading towards Stone duality.
Benoit: Yes, that’s a perfect illustration. The forgetful functor from the category of continuous maps to the category of functions is much less well-behaved than its analogue for, say, groups. A group homomorphism is in isomorphism exactly when its underlying function is an isomorphism, but a continuous map can have an isomorphism as its underlying function without being an isomorphism itself.
David: Pointless topology seems a tantalizing prospect, but for one thing. All the examples I’ve seen of Stone dualities restrict the types of topological spaces involved to, say, sober spaces. Admittedly, I haven’t seen very much of these things since most of my professors have been of the same opinion as that earlier commenter who suggested that point-set topology is basically a done deal that should be flown through as quickly as possible and left behind.
I should also make clear that I’m not asserting that these connections are unknown, but only that I don’t know them. I can see that there should be something there, but I don’t know what it is.
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