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 $\ds \bigsqcup_{i \mathop \in I} J_i$-indexed product (where $\ds \bigsqcup$ 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)