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.)
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| - Cantor cube (en)
- Cubo de Cantor (pt)
- Kostka Cantora (pl)
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| - Kostka Cantora (ciężaru gdzie jest nieskończoną liczbą kardynalną) – przestrzeń produktowa kopii zbioru z topologią dyskretną. Kostka Cantora ciężaru oznacza jest zwykle symbolem – dokładniej: gdzie jest dowolnym zbiorem mocy oraz dla każdego zbiór jest dwuelementową przestrzenią dyskretną, np. Dla przestrzeń nazywamy zbiorem Cantora. (pl)
- Em matemática, mas especificamente em topologia geral, o cubo de Cantor é a generalização do conjunto ternário de Cantor. (pt)
- 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.) (en)
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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 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. (en)
- Kostka Cantora (ciężaru gdzie jest nieskończoną liczbą kardynalną) – przestrzeń produktowa kopii zbioru z topologią dyskretną. Kostka Cantora ciężaru oznacza jest zwykle symbolem – dokładniej: gdzie jest dowolnym zbiorem mocy oraz dla każdego zbiór jest dwuelementową przestrzenią dyskretną, np. Dla przestrzeń nazywamy zbiorem Cantora. (pl)
- Em matemática, mas especificamente em topologia geral, o cubo de Cantor é a generalização do conjunto ternário de Cantor. (pt)
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