I swear the concept of compactness was invented to remedy the shortcomings of closedness. Compact sets are closed (in Hausdorff spaces and therefore metric spaces), so compactness is stricter than closedness. It evidently patches some feebleness in the definition of closedness to make it more useful.

Closedness of a set in a metric space (“includes all limit points”), by the sound of it, really wants to be something akin to “has solid boundaries.” But it isn’t. The problem is that the existence of limit points depends on the embedding space. If the embedding space lacks those limit points, then a set in it can be technically closed even though it isn’t really “like” other closed sets. For example, the set \(\mathbb R\) in space \((\mathbb R, d_{\text{Eucl.}})\) is closed, because the space has no point called \(\infty\).
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QE3 was announced today and reactions have been relatively muted. There are some complaints that money is again being redistributed from asset holders to debtors via the mechanism of negative real rates. It seems like a good occasion to put forth two oddities that I’ve always seen as embedded in capital markets as they’re currently constructed. They are: the assumption that money doesn’t spoil, and the assumption of market optimality.
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After more than 1 year — judging by this post — the Alewife bound part of Pythagoras is still broken. Up till a few months ago the graffiti stayed off. But finally somebody caught on that they aren’t actually going to fix this a second time. Blame it on shoddy mechanics — the poor handle broke after only a month in use the first time they fixed it.

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