15/11/29 08:02:05.78 SasjpBzo.net
>>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.
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nowhere dense >>52 も
URLﾘﾝｸ(en.wikipedia.org)
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.
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