Talk:Topological Product with Singleton
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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)
- 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)
- 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)
- 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)