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Relatively Compact Subset
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (since a closed subset of a compact space is compact). And in an arbitrary topological space every subset of a relatively compact set is relatively compact. Every compact subset of a Hausdorff space is relatively compact. In a non-Hausdorff space, such as the particular point topology on an infinite set, the closure of a compact subset is ''not'' necessarily compact; said differently, a compact subset of a non-Hausdorff space is not necessarily relatively compact. Every compact subset of a (possibly non-Hausdorff) topological vector space is complete and relatively compact. In the case of a metric topology, or more generally when sequences may be used to test for compactness, the criterion for relative compactness becomes that any sequence in ... [...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] |
Normal Family
In mathematics, with special application to complex analysis, a ''normal family'' is a pre-compact subset of the space of continuous functions. Informally, this means that the functions in the family are not widely spread out, but rather stick together in a somewhat "clustered" manner. Note that a compact family of continuous functions is automatically a normal family. Sometimes, if each function in a normal family ''F'' satisfies a particular property (e.g. is holomorphic), then the property also holds for each limit point of the set ''F''. More formally, let ''X'' and ''Y'' be topological spaces. The set of continuous functions f: X \to Y has a natural topology called the compact-open topology. A normal family is a pre-compact subset with respect to this topology. If ''Y'' is a metric space, then the compact-open topology is equivalent to the topology of compact convergence, and we obtain a definition which is closer to the classical one: A collection ''F'' of continuous func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Totally Bounded Space
In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size” (where the meaning of “size” depends on the structure of the ambient space). The term precompact (or pre-compact) is sometimes used with the same meaning, but precompact is also used to mean relatively compact. These definitions coincide for subsets of a complete metric space, but not in general. In metric spaces A metric space (M,d) is ''totally bounded'' if and only if for every real number \varepsilon > 0, there exists a finite collection of open balls in ''M'' of radius \varepsilon whose union contains . Equivalently, the metric space ''M'' is totally bounded if and only if for every \varepsilon >0, there exists a finite cover such that the radius of each element of the cover is at most \varepsilon. This is equivale ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compactly Embedded
In mathematics, the notion of being compactly embedded expresses the idea that one set or space is "well contained" inside another. There are versions of this concept appropriate to general topology and functional analysis. Definition (topological spaces) Let (''X'', ''T'') be a topological space, and let ''V'' and ''W'' be subsets of ''X''. We say that ''V'' is compactly embedded in ''W'', and write ''V'' ⊂⊂ ''W'', if * ''V'' ⊆ Cl(''V'') ⊆ Int(''W''), where Cl(''V'') denotes the closure of ''V'', and Int(''W'') denotes the interior of ''W''; and * Cl(''V'') is compact. Definition (normed spaces) Let ''X'' and ''Y'' be two normed vector spaces with norms , , •, , ''X'' and , , •, , ''Y'' respectively, and suppose that ''X'' ⊆ ''Y''. We say that ''X'' is compactly embedded in ''Y'', and write ''X'' ⊂⊂ ''Y'', if * ''X'' is continuously embedded in ''Y''; i.e., there is a constant ''C'' such that , , ''x'', , ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Periodic Function
In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Harald Bohr and later generalized by Vyacheslav Stepanov, Hermann Weyl and Abram Samoilovitch Besicovitch, amongst others. There is also a notion of almost periodic functions on locally compact abelian groups, first studied by John von Neumann. Almost periodicity is a property of dynamical systems that appear to retrace their paths through phase space, but not exactly. An example would be a planetary system, with planets in orbits moving with periods that are not commensurable (i.e., with a period vector that is not proportional to a vector of integers). A theorem of Kronecker from diophantine approximation can be used to show that any particular configuration that occurs once, will recur to within any specified accuracy: if we wait long eno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
