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Thus, the Cantor set is a homogeneous space in the sense that for any two points and in the Cantor set , there exists a homeomorphism : with () =. An explicit construction of h {\displaystyle h} can be described more easily if we see the Cantor set as a product space of countably many copies of the discrete space { 0 , 1 } {\displaystyle \{0,1\}} .
After black intervals have been removed, the white points which remain are a nowhere dense set of measure 1/2. In mathematics, the Smith–Volterra–Cantor set (SVC), ε-Cantor set, [1] or fat Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it contains no intervals), yet has positive measure.
Since the p-adic integers are homeomorphic to the Cantor set, they form a compact set. Any global field K is a discrete additive subgroup of its adele ring, and the quotient space is compact. This was used in John Tate's thesis to allow harmonic analysis to be used in number theory.
By the Wallis product, the area of the resulting set is π / 4 , unlike the standard Sierpiński carpet which has zero limiting area. Although the Wallis sieve has positive Lebesgue measure , no subset that is a Cartesian product of two sets of real numbers has this property, so its Jordan measure is zero.
Axiom of power set; Boolean-valued model; Burali-Forti paradox; Cantor's back-and-forth method; Cantor's diagonal argument; Cantor's first uncountability proof; Cantor's paradox; Cantor's theorem; Cantor–Bernstein–Schroeder theorem; Cardinal number. Aleph number; Beth number; Hartogs number; Cardinality; Cartesian product; Class (set theory ...
Cantor's set theory was controversial at the start, but later became largely accepted. Most modern mathematics textbooks implicitly use Cantor's views on mathematical infinity . For example, a line is generally presented as the infinite set of its points, and it is commonly taught that there are more real numbers than rational numbers (see ...
In mathematics, a Cantor algebra, named after Georg Cantor, is one of two closely related Boolean algebras, one countable and one complete. The countable Cantor algebra is the Boolean algebra of all clopen subsets of the Cantor set. This is the free Boolean algebra on a countable number of generators. Up to isomorphism, this is the only ...
Cantor restricted his first theorem to the set of real algebraic numbers even though Dedekind had sent him a proof that handled all algebraic numbers. [20] Cantor did this for expository reasons and because of "local circumstances". [53] This restriction simplifies the article because the second theorem works with real sequences.