Definition talk:Product Topology/Natural Basis

Could we add this as an equivalent definition?

$\BB = \{\prod_{i = 1}^n U_i \mid \forall i \in \{1,\ldots,n\} \ U_i \in \tau_i \}$

The equivalence follows from the fact that each element of the form $\prod_{i = 1}^n U_i$ is the intersection $\bigcap_{i = 1}^n \pr_i^{-1}(U_i)$ (since for all $k$

$\ds \pr_k^{-1}(U_k) = \left(\prod_{i = 1}^{k - 1} X_i \right) \times U_k \times \left(\prod_{i = k + 1}^{n} X_i\right) = X_1 \times \cdots \times X_{k - 1} \times U_k \times X_{k + 1} \times \cdots \times X_n$

), and the basis is generated from the sub-basis precisely by forming all interesections. --Plammens (talk) 18:43, 16 April 2021 (UTC)

As per our house rules (it says it somewhere, dunno where now, following after considerable discussion on the matter caused by a large number of unsourced and questionable definitions), as long as you can find this definition in a printed source text which expositions (is that a word?) the subject rigorously, then yes, your proposal is a good one. If it's just an equivalent definition that you came up with yourself, then I'd be less open to the suggestion. --prime mover (talk) 20:26, 16 April 2021 (UTC)

The closest source from a book I've been able to find is from Munkres' "Topology" (apologies for the screenshots, the PDF I have doesn't have text metadata, and I can't be bothered to transcribe them right now).

While it doesn't explicitly state this as a definition, it defines a generalised version of this (i.e. for possibly infinite products) as a basis for the box topology, and then states that when the product is finite, the box topology and the product topology coincide (hence we can also define this as a basis for [finite] product topology). Something similar is mentioned here (Encylopedia of Math). Wikipedia directly mentions this as a basis for the product topology (last sentence in the first paragraph), but I haven't been able to find the primary source for that.

Although to be honest, I've found these after the fact (as you might have guessed from the phrasing). My main motivation for proposing this is that this seems a much more natural (valga la reduncancia) definition of the natural basis for the product topology: it can be expressed with the tagline "the set of all products of open sets", or, to be a bit more precise, "the set of all products of sets that are open in their respective factor space". In any case, I understand why the house rules are what they are, and why you might not want to add this without a source. --Plammens (talk) 11:50, 17 April 2021 (UTC)

Ahaaa, yes. We already have the Definition:Box Topology defined. As Munkres says, the box topology and the product topology are only the same when the product is finite.

My personal view is that it would be pointless to add another definition of the Finite Product Topology as an equivalent definition, when we already have the Box Topology defined. Without checking (this is something which User:Leigh.Samphier took on board a while back. And see this: Natural Basis of Product Topology of Finite Product.

So what you say has already been covered. --prime mover (talk) 12:22, 17 April 2021 (UTC)

Doh, everything I'm saying is linked to in the "Also see" section! Completely missed that, cheers. The only thing that I find a bit strange is that Natural Basis of Product Topology is in the "flavour" of a result/theorem instead of a proof of the equivalence of two alternative definitions. (And in fact, this is not just in $\mathsf{Pr} \infty \mathsf{fWiki}$; all the sources I've seen do it like this.) I personally see $\{\prod_{i = 1}^n U_i \mid \forall i \ U_i \in \tau_i \}$ (or $\{\prod_{i} U_i \mid \forall i \ U_i \in \tau_i\,,\ \text{finitely many $i$ such that $U_i \ne X_i$} \}$ if it's countably infinite) as such an elegant and straightforward characterization that I would consider it as an alternative definition. But at this point I guess it's just a matter of taste. Or I might just be missing something. --plammens (talk) 13:34, 17 April 2021 (UTC)

Ah, I think I see why. In my mind I was only considering the product topology on finite or countable products, but like the box product, it can be defined for arbitrary index sets of factor spaces. And as Munkres and you noted, these two are generally different, but coincide in the finite case. Now I see why it makes more sense to have it as a result instead of a definition for a special case (Definition:Product Topology/Finite Product). Sorry for the spamming. --plammens (talk) 13:44, 17 April 2021 (UTC)

Sorry late to the party. The equivalent definition is indeed $\{\prod_{i} U_i \mid \forall i \ U_i \in \tau_i\,,\ \text{finitely many $i$ such that $U_i \ne X_i$} \}$. This is the subject of the theorem Natural Basis of Product Topology. So this could be offered as a second definition (I found a source: Willard) and the theorem Natural Basis of Product Topology would be the equivalence theorem.

This should be sufficient to be able to write:

Let $X,Y$ be topological spaces.

Let $U \times V$ be an open set in the natural basis of the product topology on $X \times Y$

--Leigh.Samphier (talk) 21:53, 17 April 2021 (UTC)

You could also add $\BB = \{\prod_{i = 1}^n U_i \mid \forall i \in \{1,\ldots,n\} \ U_i \in \tau_i \}$ as another definition for Natural Basis for the finite case Definition:Product Topology/Finite Product/Natural Basis. A source for this definition is Book:John M. Lee/Introduction to Topological Manifolds --Leigh.Samphier (talk) 23:04, 17 April 2021 (UTC)

I would rather that any new definitions are done by whoever has direct access to the source works than as instruction at second hand from someone else who has them. Then the person can ensure that the context is followed up, rather than just chucking definitions in because they can and because it makes us look clever. My eyes are still bleeding from the job done on piecewise continuity.

Having said that, I still wonder why it's necessary to repeat the definitions for box topology as definitions of the finite product topology, as it doesn't seem to make a great deal of sense. Because it does *not* define the product topology for infinite / uncountable / whatever product spaces, so why add a proliferation of multiple further definitions when we don't need to? Unfortunately some writers have decided to go down this ludicrously stupid route, and so we have to cater for that.

Would it be a compromise to sideline these definitions as "Also defined as" sections, to make it clear that this approach is not mainstream? --prime mover (talk) 23:15, 17 April 2021 (UTC)

I had one of those moments myself last week, when I thought everybody in the world had missed something obvious. "So why don't they define the Definition:Diameter of Conic Section as a chord which passes through the center?" It took considerable research and making an arsehole of myself on MSE before I realised the obvious reason: because such a definition is not universally applicable.

After a bit of this, you come to the conclusion that if something obvious is not declared as a mathematical definition, or theorem, or whatever, then it is not going to because all of the mathematicians in history are too foolish to have noticed it, it's because it's probably wrong for some reason.

That said, it is often profitable to take care to document this wrong obviousness and explain why, despite the fact that it's obvious, it's wrong. --prime mover (talk) 14:03, 17 April 2021 (UTC)