Definition talk:Product Topology/Finite Product

The product topology on $\XX$ is defined as the initial topology $\tau$ on $\XX$ with respect to $\family {\pr_i}_{i \mathop \in I}$:

$\tau = \set {\pr_i^{-1} \sqbrk U: i \in I, U \in \tau_i}$

Is this the actual topology or just its sub-basis? It seems we would be missing on intersections like $\pr_i^{-1} \sqbrk U \cap \pr_j^{-1} \sqbrk V$ where $U \in \tau_i$, $V \in \tau_j$, for example. --Plammens (talk) 15:15, 14 April 2021 (UTC)

All a bit of a mess. The product topology is the initial topology, which in turn is defined as the topology generated by the preimages. Hence there is an implicit "generated from a basis / subbasis" already in there.

This area was pieced together from a number of texts which were more or less comprehensible and explanatory, e.g. Sutherland takes great pains to explain what a product space is, but takes it from the context of 2 factor spaces. Steen and Seebach, on the other hand, cover it extremely compactly on the way to get somewhere else. The definition (as far as the words go) seems to be correct, but the actual symbolic presentation may need work. We also have the complication that Sutherland then goes on to define a general finite product, which is where the flow seems to have become compromised. I will revisit it in due course, but it won't be today as I have other things to do. --prime mover (talk) 15:35, 14 April 2021 (UTC)