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Universal Spaces In The Topology And Topological Dynamics
In mathematics, a universal space is a certain metric space that contains all metric spaces whose dimension is bounded by some fixed constant. A similar definition exists in topological dynamics. Definition Given a class \textstyle \mathcal of topological spaces, \textstyle \mathbb\in\mathcal is universal for \textstyle \mathcal if each member of \textstyle \mathcal embeds in \textstyle \mathbb. Menger stated and proved the case \textstyle d=1 of the following theorem. The theorem in full generality was proven by Nöbeling. Theorem: The \textstyle (2d+1)-dimensional cube \textstyle ,1 is universal for the class of compact metric spaces whose Lebesgue covering dimension is less than \textstyle d. Nöbeling went further and proved: Theorem: The subspace of \textstyle ,1 consisting of set of points, at most \textstyle d of whose coordinates are rational, is universal for the class of separable metric spaces whose Lebesgue covering dimension is less than \textstyle d. The la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
