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- Cantor_cube abstract "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.".
- Cantor_cube wikiPageID "4999981".
- Cantor_cube wikiPageLength "1757".
- Cantor_cube wikiPageOutDegree "20".
- Cantor_cube wikiPageRevisionID "672639001".
- Cantor_cube wikiPageWikiLink Absolute_extensor.
- Cantor_cube wikiPageWikiLink Cantor_space.
- Cantor_cube wikiPageWikiLink Category:Georg_Cantor.
- Cantor_cube wikiPageWikiLink Category:Topological_groups.
- Cantor_cube wikiPageWikiLink Compact_group.
- Cantor_cube wikiPageWikiLink Compact_space.
- Cantor_cube wikiPageWikiLink Continuous_function.
- Cantor_cube wikiPageWikiLink Continuous_function_(topology).
- Cantor_cube wikiPageWikiLink Countable_set.
- Cantor_cube wikiPageWikiLink Countably_infinite_set.
- Cantor_cube wikiPageWikiLink Cyclic_group.
- Cantor_cube wikiPageWikiLink Cyclic_group_of_order_2.
- Cantor_cube wikiPageWikiLink Direct_product_of_groups.
- Cantor_cube wikiPageWikiLink Discrete_space.
- Cantor_cube wikiPageWikiLink Discrete_topology.
- Cantor_cube wikiPageWikiLink Group_direct_product.
- Cantor_cube wikiPageWikiLink Hausdorff_space.
- Cantor_cube wikiPageWikiLink Homeomorphic.
- Cantor_cube wikiPageWikiLink Homeomorphism.
- Cantor_cube wikiPageWikiLink Homogeneous_space.
- Cantor_cube wikiPageWikiLink Image_(mathematics).
- Cantor_cube wikiPageWikiLink Mathematics.
- Cantor_cube wikiPageWikiLink Product_topology.
- Cantor_cube wikiPageWikiLink Topological_group.
- Cantor_cube wikiPageWikiLink Zero-dimensional_space.
- Cantor_cube wikiPageWikiLinkText "Cantor cube".
- Cantor_cube wikiPageWikiLinkText "Cantor group".
- Cantor_cube author "A.A. Mal'tsev".
- Cantor_cube hasPhotoCollection Cantor_cube.
- Cantor_cube id "C/c023230".
- Cantor_cube title "Colon".
- Cantor_cube wikiPageUsesTemplate Template:Cite_book.
- Cantor_cube wikiPageUsesTemplate Template:Springer.
- Cantor_cube subject Category:Georg_Cantor.
- Cantor_cube subject Category:Topological_groups.
- Cantor_cube hypernym Group.
- Cantor_cube type Band.
- Cantor_cube type Space.
- Cantor_cube comment "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.".
- Cantor_cube label "Cantor cube".
- Cantor_cube sameAs Kostka_Cantora.
- Cantor_cube sameAs Cubo_de_Cantor.
- Cantor_cube sameAs m.0cz6_t.
- Cantor_cube sameAs Q5034034.
- Cantor_cube sameAs Q5034034.
- Cantor_cube wasDerivedFrom Cantor_cube?oldid=672639001.
- Cantor_cube isPrimaryTopicOf Cantor_cube.