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現代数学の系譜11 ガロア理論を読む17 [転載禁止]©2ch.net

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>>53
> >>52
> つづき
> Every point of the Cantor set is also an accumulation point of the complement of the Cantor set.
> 
> For any two points in the Cantor set, there will be some ternary digit where they differ ? one will have 0 and the other 2.
> By splitting the Cantor set into "halves" depending on the value of this digit, one obtains a partition of the Cantor set into two closed sets that separate the original two points.
> In the relative topology on the Cantor set, the points have been separated by a clopen set. Consequently the Cantor set is totally disconnected. As a compact totally disconnected Hausdorff space, the Cantor set is an example of a Stone space.
> 引用おわり
> 
> nowhere dense >>52 も
> https://en.wikipedia.org/wiki/Nowhere_dense_set
> In mathematics, a nowhere dense set in a topological space is a set whose closure has empty interior.
> In a very loose sense, it is a set whose elements aren't tightly clustered close together (as defined by the topology on the space) anywhere at all.
> The order of operations is important. For example, the set of rational numbers, as a subset of R, has the property that the interior has an empty closure, but it is not nowhere dense; in fact it is dense in R.
> Equivalently, a nowhere dense set is a set that is not dense in any nonempty open set.
> 
> Nowhere dense sets with positive measure
> A nowhere dense set is not necessarily negligible in every sense.
> For example, if X is the unit interval [0,1], not only is it possible to have a dense set of Lebesgue measure zero
> (such as the set of rationals), but it is also possible to have a nowhere dense set with positive measure.
> 引用おわり

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