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Arens–Fort Space
In mathematics, the Arens–Fort space is a special example in the theory of topological spaces, named for Richard Friederich Arens and M. K. Fort, Jr. Definition The Arens–Fort space is the topological space (X,\tau) where X is the set of ordered pairs of non-negative integers (m, n). A subset U \subseteq X is open, that is, belongs to \tau, if and only if: * U does not contain (0, 0), or * U contains (0, 0) and also all but a finite number of points of all but a finite number of columns, where a column is a set \ with 0 \leq m \in \mathbb fixed. In other words, an open set is only "allowed" to contain (0, 0) if only a finite number of its columns contain significant gaps, where a gap in a column is significant if it omits an infinite number of points. Properties It is * Hausdorff * regular * normal It is not: * second-countable * first-countable * metrizable * compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
