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Mathoverflow is intimidating, so I thought I'd ask here first (second). If I don't get any useful answers here in a few days, I'll ask there.

$Q_0$: Is there any use for a topology on the (continuum of) smooth structures on $\mathbb{R}^4$?

$Q_1$: If so, is a useful topology known?

The only reference I have found is a paper by K. Kuga, "A note on Lipschitz distance for smooth structures on noncompact manifolds", MR1117158 (92f:57025). He apparently shows that several obvious topologies are discrete. One might be able to metrize the maximal exotic $\mathbb{R}^4$ and use some variant of the Hausdorff-Gromov pseudometric, but that's such an obvious idea that I'd be surprised if it works, given that I haven't seen it.

Edit 2: The previous edit was edited into the comments below. Edit 3: I may as well clarify "is there any use?" in the OP as the zeroth question.

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    @Zev: Thank you! I'm registering now. We'll see if this works!2011-09-18

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