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In point-set topology, one always learns about the box topology: the topology on an infinite product $X = \prod_{i \in I} X_i$ generated by sets of the form $U = \prod_{i \in I} U_i$, where $U_i \subset X_i$ is open. This seems naively like a "good" topology to use for $X$. However, one quickly learns that this is not so; that the product topology is the natural one.

The box topology has many strange properties that make it a good source for counterexamples, but I am not aware of it having any other applications. So I would like to know:

Are there examples of using the box topology to prove interesting "positive" statements?

Edit: And to pursue a comment of Jim Conant:

Are there "non-artificial" problems where the box topology arises naturally?

Edit: The title is perhaps too flippant. I don't mean to minimize the obvious significance of the box topology as a counterexample. However, for the purposes of this question I am interested in positive results. I'm not looking to be convinced that counterexamples are useful; I know that they are.

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    Being a good source for counterexamples is a very good thing indeed. It helps us in our attempt to understand where, how and why certain properties break in the general case.2011-05-02
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    Kunen-Vaughan, *Handbook of set-theoretic topology*, Chapter 4 (written by S.P. Williams) contains a lengthy survey on box products. The results there seem to be more of the nature that Asaf points out.2011-05-02
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    I like this question. For example, are there any cases where a topological space is given from some other context, and it turns out that it is homeomorphic to a box product?2011-05-02
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    There is something called the "Box product problem", which asks whether the box product of countably many copies of the real line is normal; it spurred some interest in the 70s, at least, with Mary Ellen Rudin proving that the Continuum Hypothesis implies this is the case, and E.K. van Douwen proving that in general the box product of normal spaces need not be normal. From a quick look through "Counterexamples in Topology" and a quick google search, problems involving the box topology (in the infinite index set) seem to be closely connected to problems of set-theory (CH, AC, etc).2011-05-02
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    @Arturo: I believe that was one of the things that sparked general topology as it is today.2011-05-02
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    Based on examples in Munkres, I would think the box topology could potentially be used to prove certain theorems about uniform convergence topologies.2011-12-10
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    @dfeuer: Could you elaborate?2011-12-13
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    For countable products of separable metric spaces, the Borel sets of the product topology and the box topology coincide, so maybe this angle provides some use.2012-05-31

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