Editing Cantor set (section)

=== Topological and analytical properties ===

Although "the" Cantor set typically refers to the original, middle-thirds Cantor set described above, topologists often talk about "a" Cantor set, which means any [[topological space]] that is [[homeomorphic]] (topologically equivalent) to it.

As the above summation argument shows, the Cantor set is uncountable but has [[Lebesgue measure]] 0. Since the Cantor set is the [[complement (set theory)|complement]] of a [[union (set theory)|union]] of [[open set]]s, it itself is a [[closed set|closed]] subset of the reals, and therefore a [[complete metric space]]. Since it is also [[totally bounded]], the [[Heine–Borel theorem]] says that it must be [[compact space|compact]].

For any point in the Cantor set and any arbitrarily small [[neighborhood (mathematics)|neighborhood]] of the point, there is some other number with a ternary numeral of only 0s and 2s, as well as numbers whose ternary numerals contain 1s.  Hence, every point in the Cantor set is an [[accumulation point]] (also called a cluster point or limit point) of the Cantor set, but none is an [[interior point]]. A closed set in which every point is an accumulation point is also called a [[perfect set]] in [[topology]], while a closed subset of the interval with no interior points is [[Nowhere dense set|nowhere dense]] in the interval.

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]].

As a topological space, the Cantor set is naturally [[Homeomorphism|homeomorphic]] to the [[product topology|product]] of countably many copies of the space <math>\{0, 1\}</math>, where each copy carries the [[discrete topology]]. This is the space of all [[sequence]]s in two digits 
:<math>2^\mathbb{N} = \{(x_n) \mid x_n \in \{0,1\} \text{ for } n \in \mathbb{N}\},</math>

which can also be identified with the set of [[p-adic integer|2-adic integers]]. The [[basis (topology)|basis]] for the open sets of the [[product topology]] are [[cylinder set]]s; the homeomorphism maps these to the [[subspace topology]] that the Cantor set inherits from the natural topology on the [[real line]]. This characterization of the [[Cantor space]] as a product of compact spaces gives a second proof that Cantor space is compact, via [[Tychonoff's theorem]].

From the above characterization, the Cantor set is [[Homeomorphism|homeomorphic]] to the [[p-adic integer|''p''-adic integers]], and, if one point is removed from it, to the [[p-adic number|''p''-adic numbers]].

The Cantor set is a subset of the reals, which are a [[metric space]] with respect to the [[absolute difference|ordinary distance metric]]; therefore the Cantor set itself is a metric space, by using that same metric. Alternatively, one can use the [[p-adic metric|''p''-adic metric]] on <math>2^\mathbb{N}</math>: given two sequences <math>(x_n),(y_n)\in 2^\mathbb{N}</math>, the distance between them is <math>d((x_n),(y_n)) = 2^{-k}</math>, where <math>k</math> is the smallest index such that <math>x_k \ne y_k</math>; if there is no such index, then the two sequences are the same, and one defines the distance to be zero. These two metrics generate the same [[topological space|topology]] on the Cantor set.

We have seen above that the Cantor set is a totally disconnected [[perfect set|perfect]] compact metric space. Indeed, in a sense it is the only one: every nonempty totally disconnected perfect compact metric space is [[Homeomorphism|homeomorphic]] to the Cantor set. See [[Cantor space]] for more on spaces [[Homeomorphism|homeomorphic]] to the Cantor set.

The Cantor set is sometimes regarded as "universal" in the [[category (mathematics)|category]] of [[compact space|compact]] metric spaces, since any compact metric space is a [[continuous function (topology)|continuous]] [[image (mathematics)|image]] of the Cantor set; however this construction is not unique and so the Cantor set is not [[universal property|universal]] in the precise [[category theory|categorical]] sense. The "universal" property has important applications in [[functional analysis]], where it is sometimes known as the ''representation theorem for compact metric spaces''.<ref>{{cite book | first=Stephen | last=Willard | title=General Topology | publisher=Addison-Wesley | date=1968 | asin=B0000EG7Q0}}</ref>

For any [[integer]] ''q'' ≥ 2, the topology on the [[group (mathematics)|group]] G = '''Z'''<sub>''q''</sub><sup>ω</sup> (the countable direct sum) is discrete.<!-- I don't know how the Z_q^w should be LaTeX-ified. --> Although the [[Pontrjagin dual]] Γ is also '''Z'''<sub>''q''</sub><sup>ω</sup>, the topology of Γ is compact. One can see that Γ is totally disconnected and perfect - thus it is [[Homeomorphism|homeomorphic]] to the Cantor set. It is easiest to write out the homeomorphism explicitly in the case ''q'' = 2. (See Rudin 1962 p 40.)