|
Point-finite Collection
In mathematics, a collection \mathcal of subsets of a topological space X is said to be point-finite if every point of X lies in only finitely many members of \mathcal.. A topological space in which every open cover admits a point-finite open refinement is called metacompact. Every locally finite collection of subsets of a topological space is also point-finite. A topological space in which every open cover admits a locally finite open refinement is called paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, .... Every paracompact space is therefore metacompact. References General topology {{topology-stub ... [...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] |
Topological Space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\subset X, then C is a cover of X if \bigcup_U_ = X. Thus the collection \lbrace U_\alpha : \alpha \in A \rbrace is a cover of X if each element of X belongs to at least one of the subsets U_. Cover in topology Covers are commonly used in the context of topology. If the set X is a topological space, then a ''cover'' C of X is a collection of subsets \_ of X whose union is the whole space X. In this case we say that C ''covers'' X, or that the sets U_\alpha ''cover'' X. Also, if Y is a (topological) subspace of X, then a ''cover'' of Y is a collection of subsets C=\_ of X whose union contains Y, i.e., C is a cover of Y if :Y \subseteq \bigcup_U_. That is, we may cover Y with either open sets in Y itself, or cover Y by open sets in the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Refinement (topology)
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\subset X, then C is a cover of X if \bigcup_U_ = X. Thus the collection \lbrace U_\alpha : \alpha \in A \rbrace is a cover of X if each element of X belongs to at least one of the subsets U_. Cover in topology Covers are commonly used in the context of topology. If the set X is a topological space, then a ''cover'' C of X is a collection of subsets \_ of X whose union is the whole space X. In this case we say that C ''covers'' X, or that the sets U_\alpha ''cover'' X. Also, if Y is a (topological) subspace of X, then a ''cover'' of Y is a collection of subsets C=\_ of X whose union contains Y, i.e., C is a cover of Y if :Y \subseteq \bigcup_U_. That is, we may cover Y with either open sets in Y itself, or cover Y by open sets in the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metacompact Space
In the mathematical field of general topology, a topological space is said to be metacompact if every open cover has a point-finite open refinement. That is, given any open cover of the topological space, there is a refinement that is again an open cover with the property that every point is contained only in finitely many sets of the refining cover. A space is countably metacompact if every countable open cover has a point-finite open refinement. Properties The following can be said about metacompactness in relation to other properties of topological spaces: * Every paracompact space is metacompact. This implies that every compact space is metacompact, and every metric space is metacompact. The converse does not hold: a counter-example is the Dieudonné plank. * Every metacompact space is orthocompact. * Every metacompact normal space is a shrinking space * The product of a compact space and a metacompact space is metacompact. This follows from the tube lemma. * An easy exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Finite Collection
In the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space. It is fundamental in the study of paracompactness and topological dimension. A collection of subsets of a topological space X is said to be locally finite if each point in the space has a neighbourhood that intersects only finitely many of the sets in the collection. Note that the term locally finite has different meanings in other mathematical fields. Examples and properties A finite collection of subsets of a topological space is locally finite. Infinite collections can also be locally finite: for example, the collection of all subsets of \mathbb of the form (n, n+2) for an integer n. A countable collection of subsets need not be locally finite, as shown by the collection of all subsets of \mathbb of the form (-n, n) for a natural number ''n''. If a collection of sets is locally finite, the collection of all closures of these sets is also locally finite. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paracompact Space
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff. Every closed subspace of a paracompact space is paracompact. While compact subsets of Hausdorff spaces are always closed, this is not true for paracompact subsets. A space such that every subspace of it is a paracompact space is called hereditarily paracompact. This is equivalent to requiring that every open subspace be paracompact. Tychonoff's theorem (which states that the product of any collection of compact topological spaces is compact) does not generalize to paracompact spaces in that the product of paracompact spaces need not be paracompact. Howeve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
