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In mathematics, a Cantor cube is a topological group of the form {0, 1}A for some index set A. Its algebraic and topological structures are the group direct product and product topology over the cyclic group of order 2 (which is itself given the discrete topology).
If A is a countably infinite set, the corresponding Cantor cube is a Cantor space. Cantor cubes are special among compact groups because every compact group is a continuous image of one, although usually not a homomorphic image. (The literature can be unclear, so for safety, assume all spaces are Hausdorff.)
Topologically, any Cantor cube is:
homogeneous;
compact;
zero-dimensional;
AE(0), an absolute extensor for compact zero-dimensional spaces. (Every map from a closed subset of such a space into a Cantor cube extends to the whole space.)
By a theorem of Schepin, these four properties characterize Cantor cubes; any space satisfying the properties is homeomorphic to a Cantor cube.
In fact, every AE(0) space is the continuous image of a Cantor cube, and with some effort one can prove that every compact group is AE(0). It follows that every zero-dimensional compact group is homeomorphic to a Cantor cube, and every compact group is a continuous image of a Cantor cube.
In mathematics, a Cantorcube is a topological group of the form {0, 1}A for some index set A. Its algebraic and topological structures are the group...
The indicator function of the Cantor set Smith–Volterra–Cantor set Cantor function Cantorcube Antoine's necklace Koch snowflake Knaster–Kuratowski fan...
intersection of the sponge with any diagonal of the cube or any midline of the faces is a Cantor set. The cross-section of the sponge through its centroid...
mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it...
after Georg Cantor (1845–1918), a German mathematician. Cantor algebra CantorcubeCantor distribution Cantor function Cantor normal form Cantor pairing function...
space Cantor set, Cantor space, Cantorcube Space-filling curve Topologist's sine curve Uniform norm Weak topology Strong topology Hilbert cube Lower...
choice by constructing a continuous function from the usual Cantor set onto the Hilbert cube. In ℓ 2 , {\displaystyle \ell _{2},} no point has a compact...
I.; Balasoiu, M.; Osipov, V.A. (2010). "The scattering from generalized Cantor fractals". J. Appl. Crystallogr. 43 (4): 790–7. arXiv:0911.2497. doi:10...
{\mathcal {A}}} as a topological space by considering it a subset of the Cantorcube D X {\displaystyle D^{X}} , and in this case, we denote it K ( A ) {\displaystyle...
examples in general topology, a field of mathematics. Alexandrov topology Cantor space Co-kappa topology Cocountable topology Cofinite topology Compact-open...
cardinality of the set R of all real numbers. For proofs, see Cantor's diagonal argument or Cantor's first uncountability proof. If |A| ≤ |B| and |B| ≤ |A|,...
tori that are not removed at some stage, an embedding of the sphere with a Cantor set removed results. This embedding extends to a continuous map from the...
therefore are null sets when considered as subsets of the real numbers. The Cantor set is an example of an uncountable null set.[further explanation needed]...
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and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and...
number (7) and the fourth composite number (9). 8 is the first non-unitary cube prime of the form p3. With proper divisors 1, 2, and 4, it is the third power...
some cases are periodic), and in many other ways. As a consequence of Cantor's proof that the real numbers are uncountable and the rationals countable...
Since w = 1, we exit the while loop. Since c ≠ 1, it must be a perfect cube. The cube root of c, obtained by replacing x3 by x is x2 + 1, and calling the...
examples of Polish spaces are the real line, any separable Banach space, the Cantor space, and the Baire space. Additionally, some spaces that are not complete...
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space N {\displaystyle {\mathcal {N}}} , the Cantor space C {\displaystyle {\mathcal {C}}} , and the Hilbert cube I N {\displaystyle I^{\mathbb {N} }} . The...
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