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two years later and only now do i understand why the product topology is defined the way it is for infinite products
this is the right idea.
let's fix some notation. I is an indexing set, with a topological space X_i for each i in I. I'll denote the product to be X.
1/n https://twitter.com/Category_Fury/status/1263898253843509248?s=20
let's fix some notation. I is an indexing set, with a topological space X_i for each i in I. I'll denote the product to be X.
1/n https://twitter.com/Category_Fury/status/1263898253843509248?s=20
One might naively try to define the topology on X to be products of sets U_i where each U_i is open in X_i. It turns out this is "too fine", ie there are too many open sets. I learned this naive product topology as the "box topology."
2/n
2/n
(maybe because you can think of the infinite products of open sets as boxes idk)
3/n
3/n
The right definition for the product topology, though, is to take as your basis products of open sets U_i *where all but finitely many U_i are the whole space X_i*.
So if I is finite, the two definitions agree.
4/n
So if I is finite, the two definitions agree.
4/n
Thinking categorically there are things we demand out of a product. First you want projections p_i: X -->X_i, and you also want a universal property where if you have some space Y and a map f_i: Y --> X_i,
there exists uniquely a (continuous!) map f:Y -->X.
5/n
there exists uniquely a (continuous!) map f:Y -->X.
5/n
Consider the following example: Our indexing set is the natural numbers N, and each X_i is a copy of the real numbers R.
Now, in the above notation let Y also be R, and f_i: Y --> X_i be the identity R --> R.
It is clear what the map f: R --> R^N should look like...
6/n
Now, in the above notation let Y also be R, and f_i: Y --> X_i be the identity R --> R.
It is clear what the map f: R --> R^N should look like...
6/n
... as a map of sets, at least. It should send x to the infinite tuple (x,x,x,...)
However, if you give R^N the box topology, this is not continuous (it is an illuminating exercise as to why).
7/n
However, if you give R^N the box topology, this is not continuous (it is an illuminating exercise as to why).
7/n
It is continuous in the product topology though! So it's at least a useful definition. And first-year me left it at that, but it was still kind of a mystery as to *why* the product topology was defined that way.
8/n
8/n
This is why thinking categorically is so helpful. The thing that makes it all work is that *you want the projection maps to induce the topology on X*
9/n
9/n
What I mean is that you give X the coarsest topology such that all the projection maps are continuous. By coarsest I mean fewest open sets.
10/n
10/n
If you have an open set U_i in X_i, the inverse image under the projection p_i is the product of all the remaining X_j, along with U_i. This gives you a sub-basis. You take finite intersections and arbitrary unions of these to get the topology on X.
11/n
11/n
That's the key! Finite intersections. That's why you don't want an open set which is a product of infinitely many nontrivial opens, because it's more than what the projection maps demand to be continuous.
12/n, n = 12
12/n, n = 12
idk how to conclude threads like this, just kind of a fun thing i was thinking about.
by the way, thinking this way also tells you why the coproduct (or direct sum) works the way it does in a category of R-modules.
by the way, thinking this way also tells you why the coproduct (or direct sum) works the way it does in a category of R-modules.
