2012/09/27
compactness
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 \(\infty\).
(Read the article)
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