Talk:Topological Product with Singleton

Ways this could go
I added one generalization (which I will use to prove that general products of connected spaces are connected), but there may be a better way: prove that the product of two products is homeomorphic to the combined product. Dfeuer (talk) 01:52, 6 December 2012 (UTC)
 * Oh, and then do something with that.Dfeuer (talk) 01:54, 6 December 2012 (UTC)


 * Generalization is valuable even if the other result is also to be established (the latter is in fact categorical in nature, but I won't bother you with that). --Lord_Farin (talk) 09:18, 6 December 2012 (UTC)


 * You're welcome to bother me with categorical matters if you're willing to explain them in the sort of baby-talk I can understand. --Dfeuer (talk) 09:38, 6 December 2012 (UTC)


 * What I was conveying is (essentially) that an $I$-indexed product of $J_i$-indexed products is categorically (i.e. with only use of the categorical product properties) immediately a $\displaystyle \bigsqcup_{i \in I} J_i$-indexed product (where $\bigsqcup$ is the PW notation for disjoint union). It therefore does hold immediately for all categories with products (such as e.g. $\mathbf{Top}$, the category of top. spaces). --Lord_Farin (talk) 09:58, 6 December 2012 (UTC)


 * Can the categorical approach be used to prove the general form you give without dragging in lots of other categorical concepts, or would it be best to give a direct proof (as well)? --Dfeuer (talk) 19:55, 7 December 2012 (UTC)


 * I would advise a direct proof as well; currently, there is not really a description of the general product having the simplicity necessary to be able to comfortably rely on it (presently, one needs the notion of a limit, which involves a lot of preliminaries if you chase them down). --Lord_Farin (talk) 22:12, 7 December 2012 (UTC)