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).
 * Munkres - Topology - Box Topology.png
 * Munkres - Topology - Comparison of Box Topology and Product Topology.png
 * 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 ; all the sources I've seen do it like this.) I personally see $\{\prod_{i = 1}^n U_i \mid \forall i \ U_i \in \tau_i \}$ 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.