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Stream: learning: questions

Topic: continuous functions as functors?

David Egolf (Aug 24 2022 at 22:24):

I learned about the concept of "homomorphism" first in the setting of groups. The idea is that they are maps that "preserve the structure". For example, if $\phi$ is a map between additive groups, then for $\phi$ to be a homomorphism of additive groups we require $\phi(a+b) = \phi(a) + \phi(b)$ for all group elements $a,b$. This condition on preservation of structure is very similar to what we require for functors, to my understanding.

However, I've also learned that continuous maps are the homomorphisms for topological spaces. I was trying to see how we can express that a function is continuous using a similar "structure preserving" requirement as for group homomorphisms.

Let $\mathbb{N} \xrightarrow{s} \mathbb{R}$ be a sequence that converges to $S$.
Then, a function $f$ is continuous at $S$ if $\lim_{n \to \infty}f(s(n)) = f(S)$, I think.

Now define a function $L$ that takes any convergent sequence and returns its limit. Letting $C \subseteq \mathbb{R}^\mathbb{N}$ be the set of real-valued convergent sequences, then $L: C \to \mathbb{R}$.
Then $L(s) = S$. Under this notation, we have that $f$ is continuous at $S$ if $L(f(s)) = f(L(s))$.
This is similar to the structure-preserving requirement for groups; if we view $+$ as a function that takes two inputs, the structure preserving requirement becomes: $\phi(+(a,b)) = +(\phi(a), \phi(b))$. It seems like, from this perspective, a continuous functions really is "structure-preserving", where the structure is created by the operation of taking limits. If this is right, it seems cool!

As a sort of follow-up - Is there some way to view continuous functions between topological spaces as functors, from this perspective?

Edit: This was also making me wonder if one could try and define a topological space as a category where you can compute limits of sequences. This led me to discovering this interesting page: https://ncatlab.org/nlab/show/convergence+space . Neat!

Zhen Lin Low (Aug 24 2022 at 22:27):

"Homomorphism" and "homeomorphism" are different words...

David Egolf (Aug 24 2022 at 22:30):

Zhen Lin Low said:

"Homomorphism" and "homeomorphism" are different words...

Good point, thanks. I will fix it.

Max New (Aug 24 2022 at 22:57):

So your definition doesn't work for all topological spaces. The ones where it does work are https://ncatlab.org/nlab/show/subsequential+space

David Egolf (Aug 24 2022 at 23:03):

Max New said:

So your definition doesn't work for all topological spaces. The ones where it does work are https://ncatlab.org/nlab/show/subsequential+space

It would be great if you could be a little more specific (which definition? doesn't work how?). I don't really know much topology, and was basically just thinking about real numbers when I wrote this out. But thanks for the interesting link!

John Baez (Aug 24 2022 at 23:03):

There are lots of topological spaces where the topology is not determined by the collection of sequences that converge.

John Baez (Aug 24 2022 at 23:05):

This is why people invented nets... and I think the link here will help explain the issues.

John Baez (Aug 24 2022 at 23:06):

When I teach analysis, I list 3 facts about sequences that are true for metric spaces but not general topological spaces. Later I show how all these facts can be "saved" if we use nets instead of sequences.

The link mentions just one of these facts, under "Motivation". That's the one were just talking about:

1) a function between metric spaces is continuous iff it maps convergent sequences to convergent sequences.

John Baez (Aug 24 2022 at 23:08):

Here are the two other facts: for a metric space

2) a subset U is open iff given x $\in$ U and a sequence converging to x, the sequence is eventually in U;

3) a subset K is compact iff every sequence in K has a convergent subsequence.

John Baez (Aug 24 2022 at 23:15):

It's good to know how all three of these facts fail for general topological spaces, and how to save them using nets.

David Egolf (Aug 24 2022 at 23:17):

Ok, that makes a lot of sense! So in a metric space we have continuous functions interacting nicely with convergent seequences, but that's not the case in every topological space. And apparently this niceness can be recovered by using "nets", which are functions from directed sets (instead of from just the natural numbers).
That explains why my idea above won't work for all topological spaces, at least when working with sequences.
Thanks!

John Baez (Aug 24 2022 at 23:18):

Right. Basically the problem is that for really "big" topological spaces, sequences just aren't long enough. They're infinitely long... but that's only the smallest infinity!

David Egolf (Aug 24 2022 at 23:21):

Right, our sequences only have countably many elements. But one could consider a map from the ordered real numbers, for example, to get an image with many more points (uncountably many).
Also, I seem to recall that directed sets can have more complicated shapes - some kinds of "branching", I think? I don't know if that is important to the "niceness" of nets.

John Baez (Aug 24 2022 at 23:22):

Right. We need to let nets be longer than sequences... but while we're at it, we might as well let them be "wider".

John Baez (Aug 24 2022 at 23:23):

That is, we index a net by a directed set, and a directed set doesn't need to be linearly ordered.

John Baez (Aug 24 2022 at 23:24):

That lets us do tricks like this. Take as our directed set the collection of all open sets $U$ containing a point $x$, ordered by reverse inclusion so $U \ge V$ iff $U \subseteq V$. Then we can take a net indexed by this directed set!

John Baez (Aug 24 2022 at 23:26):

This allows really slick proofs.

David Egolf (Aug 24 2022 at 23:26):

That's a really cool example! It sounds like that would let you talk about approaching a point from "all sides at once", or without committing to a single path of approach.

John Baez (Aug 24 2022 at 23:27):

Exactly! In metric spaces we often say stuff like "choose a sequence $x_n$ such that $d(x_n , x) \le 1/n$".

But with nets we can say "choose a net $x_U$ such that $x \in U$", where $U$ is any open set containing $x$.

John Baez (Aug 24 2022 at 23:28):

So once you get used to them they are very easy to work with.

John Baez (Aug 24 2022 at 23:29):

The point is that open neighborhoods, rather than distances, are how we measure "closeness" in a topological space.