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Order Complex
In mathematics, the poset topology associated to a poset (''S'', ≤) is the Alexandrov topology (open sets are upper sets) on the poset of finite chains of (''S'', ≤), ordered by inclusion. Let ''V'' be a set of vertices. An abstract simplicial complex Δ is a set of finite sets of vertices, known as faces \sigma \subseteq V, such that ::\forall \rho \, \forall \sigma \!: \ \rho \subseteq \sigma \in \Delta \Rightarrow \rho \in \Delta. Given a simplicial complex Δ as above, we define a (point set) topology on Δ by declaring a subset \Gamma \subseteq \Delta be closed if and only if Γ is a simplicial complex, i.e. ::\forall \rho \, \forall \sigma \!: \ \rho \subseteq \sigma \in \Gamma \Rightarrow \rho \in \Gamma. This is the Alexandrov topology on the poset of faces of Δ. The order complex associated to a poset (''S'', ≤) has the set ''S'' as vertices, and the finite chains of (''S'', ≤) as faces. The poset topology associated to a poset (''S'', ≤) is then the Alexandro ... [...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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
