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Def: The product topology is generated by sets of the form $\prod\limits_{n\in\mathbb{N}} U_n$ where each $U_n$ is open in $X_n$ and, for all but finitely many $n$, we have $U_n = X_n$.

I am looking for an answer that explains how the open sets in the final topology is affected by the definition especially for all but finitely many $n$, we have $U_n = X_n$.

Martin Sleziak
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user2277550
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    What do you mean by "affected"? I don't understand what it is you want to know. – anon Dec 17 '16 at 16:39
  • I think OP is asking for an explanation of how the product topology differs from the box topology. For finite products they are identical, so the difference is rather subtle. – MJD Dec 17 '16 at 19:08
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    @arctictern I was having difficulty understanding what the 'for all but finitely many n, we have Un=Xn' means. So I was hoping someone would explain how the open sets in the product topology are defined ? – user2277550 Dec 17 '16 at 19:11
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    It means exactly what it says. The basic open sets are of the form $\prod U_n$ where all of the $U_n$ are the whole space $X_n$, except for finitely many exceptions where $U_n$ is a proper open subset of $X_n$. For example, if $U$ is open in $X_1$, then $$U\times X_2\times X_3\times X_4\times\cdots$$ is an open subset of $\prod X_n$ because all but one of its "factors" is the whole space in that component. Or, if $V\subseteq X_2,W\subseteq X_4$ are open subsets then $$X_1\times V\times X_3\times W\times\cdots$$ is an open subset of $\prod X_n$ as all but two of its factors are $X_n$. – anon Dec 17 '16 at 20:43
  • If it helps, you can alternatively think of the open subsets of $\prod X_n$ as being of the form $U\times \prod_{n> m}X_n\subseteq \prod_{n\ge1} X_n$ where $U$ is an open subset of $\prod_{n=1}^m X_n$ for some $m\ge1$. – anon Dec 17 '16 at 20:45
  • Also a search for "product box topology" produces several other times that similar questions have been asked. The box topology is the same as the product topology except for the finiteness condition that is puzzling you. By comparing the two constructions you may be able to understand the effect of the finiteness condition. – MJD Dec 17 '16 at 23:36
  • This answer about the convergence of sequences in the two topologies may also be useful. – MJD Dec 17 '16 at 23:49
  • I doubt the usefulness of framing an answer in terms of the difference between box and product topologies for this user. The same user asks this Question out of frustration in trying to understand essentially the same point of the definition in this previous Question. – hardmath Dec 19 '16 at 17:32

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