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)