The term finitemathematics refers either to discrete mathematics , or to a course conventionally required of business students, in which the curriculum brings together several mathematical topics, including basic probability theory , an introduction to linear programming , some theory of matrix mathematics matrices and determinants, and sometimes an abbreviated account of calculus . disambig Category Mathematical disambiguation ... more details
wiktionary Finite is the opposite of infinite . It may refer to A finite number or value that is, a real number or complex with finite modulus?? Finite set , having a number of elements given by some natural number Finite verb , being inflected for person and for tense disambig de Endlichkeit fr Fini ko it Finito simple Finite ... more details
The term locally finite has a number of different meanings in mathematics Locally finite collection of sets in a topological space Locally finite group Locally finite measure Locally finite poset Locally finite variety in the sense of universal algebra disambig ... more details
Noref date May 2010 In mathematics , a K finite function is a type of generalized trigonometric polynomial . Here K is some compact group , and the generalization is from the circle group T . From an abstract point of view, the characterization of trigonometric polynomials amongst other functions F , in the harmonic analysis of the circle, is that for functions F in any of the typical function space s, F is a trigonometric polynomial if and only if its Fourier coefficient s a sub n sub vanish for n large enough, and that this in turn is equivalent to the statement that all the translates F t by a fixed angle lie in a finite dimensional subspace. One implication here is trivial, and the other, starting from a finite dimensional invariant subspace , follows from complete reducibility of representations of T . From this formulation, the general definition can be seen for a representation of K on a vector space V , a K finite vector v in V is one for which the k . v for k in K span a finite dimensional subspace. The union of all finite dimension K invariant subspaces is itself a subspace, and K invariant, and consists of all the K finite vectors. When all v are K finite, the representation itself is called K finite. Category Representation theory of groups Reference Lectures on Lie Groups ans Lie Algebras by Roger Carter, Graeme Segal and Ian Macdonald ... more details
In algebraic geometry , a branch of mathematics , a morphism math f X rightarrow Y math of scheme theory scheme s is a finite morphism , if math Y math has an open cover by affine schemes math V i mbox ... of finite type There is another finiteness condition on morphisms of schemes, morphisms of finite type , which is much weaker than being finite. Morally, a morphism of finite type corresponds to a set ... Z rightarrow mathbb Z x, y, z langle y 3 x 4 z rangle math . This is an example of a morphism of finite ... of rings math B i rightarrow A ij math . The morphism f is called locally of finite type , if math ... the open cover math f 1 V i bigcup j U ij math can be chosen to be finite, then f is called of finite type . For example, if math k math is a field mathematics field , the scheme math mathbb A n k math has a natural morphism to math text Spec k math induced by the inclusion of Ring mathematics rings math k to k X 1, ldots,X n . math This is a morphism of finite type, but if math n 0 math then it is not a finite morphism. On the other hand, if we take the affine scheme math mbox Spec ... homomorphism math k X to k X,Y langle Y 2 X 3 X rangle. math Then this morphism is a finite morphism. Properties of finite morphisms In the following, f X Y denotes a finite morphism. The composition of two finite maps is finite. Any Grothendieck s relative point of view base change of a finite morphism is finite, i.e. if math g Z rightarrow Y math is another arbitrary morphism, then the canonical morphism math X times Y Z rightarrow Z math is finite. This corresponds to the following algebraic ... module. Closed immersion s are finite, as they are locally given by math A rightarrow A I math , where I is the Ideal ring theory ideal corresponding to the closed subscheme. Finite morphisms are closed ... of Cohen Seidenberg. Finite morphisms have finite fibres i.e. they are quasi finite morphism quasi finite . This follows from the fact that any finite k algebra, for any field k is an Artinian ... more details
In mathematics, a family of sets family math mathcal F math of Set mathematics sets is of finite character provided it has the following properties For each math A in mathcal F math , every finite set finite subset of math A math belongs to math mathcal F math . If every finite subset of a given set math A math belongs to math mathcal F math , then math A math belongs to math mathcal F math . Properties A family math mathcal F math of sets of finite character enjoys the following properties For each math A in mathcal F math , every finite or infinite subset of math A math belongs to math mathcal F math . Tukey s lemma In math mathcal F math , partial order partially ordered by inclusion, the Union set theory union of every Total order Chains chain of elements of math mathcal F math also belong to math mathcal F math , therefore, by Zorn Lemma Zorn s lemma , math mathcal F math contains at least one maximal element. Example Let V be a vector space , and let F be the family of linearly independent subsets of V . Then F is a family of finite character because a subset X &sube V is linearly dependent iff X has a finite subset which is linearly dependent . Therefore, in every vector space , there exists a maximal family of linearly independent elements. As a maximal family is a vector basis , every vector space has a possibly infinite vector basis. PlanetMath attribution id 3692 title finite character Mathlogic stub Category Set families eo Ara sistemo de finia speco fr Ensemble de parties de caract re fini ... more details
In mathematics , a finite set is a Set mathematics set that has a finite number of element mathematics elements . For example, math 2,4,6,8,10 , math is a finite set with five elements. The number of elements of a finite set is a natural number non negative integer , and is called the cardinality of the set. A set that is not finite is called infinite . For example, the set of all positive integers is infinite math 1,2,3, ldots . math Finite sets are particularly important in combinatorics , the mathematical study of counting . Many arguments involving finite sets rely on the pigeonhole principle , which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set S is called finite if there exists a bijection ... of the set, and is denoted S . Note that the empty set is considered finite, with cardinality zero. If a set is finite, its elements may be written as a sequence math S x 1,x 2, ldots,x n . math In combinatorics , a finite set with n elements is sometimes called an n set and a subset with k elements is called a k subset . For example, the set 5,6,7 is a 3 set, a finite set with three elements, and 6,7 is a 2 subset of it. Basic properties Any proper subset of a finite set S is finite and has fewer elements than S itself. As a consequence, there cannot exist a bijection between a finite set S and a proper subset of S . Any set with this property is called Dedekind finite . Using the standard Zermelo Fraenkel set theory ZFC axioms for set theory , every Dedekind finite set is also finite, but this requires the axiom of choice or at least the axiom of dependent choice . Any injective function between two finite sets of the same cardinality is also a surjective function surjection , and similarly any surjection between two finite sets of the same cardinality is also an injection. The union set theory union of two finite sets is finite, with math S cup T le S T . math In fact ... more details
In mathematics , more specifically abstract algebra , a finite ring is a ring mathematics ring not necessarily with a multiplicative identity that has a finite number of elements. Every finite field is an example of a finite ring, and the additive part of every finite ring is an example of an abelian group abelian finite group , but the concept of finite rings in their own right has a more recent history ... notes see reference . The occasion of non commutativity in finite rings was described in 1968 in the same journal 75 512&ndash 14 by K. Eldrige in two theorems If the order m of a finite ring with 1 ... non commutative finite ring with 1 has the order of a prime cubed, then the ring is isomorphic to the upper ... references in the topic of finite rings, such as Robert Ballieu 1947 Anneaux finis in Ann ... review MR0022841 of Ballieu. These are a few of the facts that are known about the number of finite rings of a given order suppose p and q represent distinct prime numbers There are two finite rings of order p . There are four finite rings of order pq . There are eleven finite rings of order p sup 2 sup . There are twenty two finite rings of order p sup 2 sup q . There are fifty two finite rings of order eight. There are 3 p 50 finite rings of order p sup 3 sup , p 2 ... s little theorem Artin Wedderburn theorem There are other deep aspects to the theory of finite rings ... s little theorem , asserts that any finite division ring is necessarily commutative and therefore a finite field . Nathan Jacobson later discovered yet another condition which guarantees commutativity ... of finite simple ring s is relatively straightforward in nature. More specifically, any finite simple ring is isomorphic to the ring math M n mathbb F q math of n by n matrices over a finite field ... is Wedderburn s little theorem . On the other hand, the classification of finite simple groups was one of the major breakthroughs of twentieth century mathematics, its proof spanning thousands ... more details
cleanup rewrite date September 2011 Groups In mathematics and abstract algebra , a finite group is a group mathematics group whose underlying set mathematics set G has finite set finite ly many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local analysis local theory of finite groups, and the theory of solvable group s and nilpotent group s. A complete determination of the structure of all finite groups is too much to hope for the number of possible structures soon becomes overwhelming. However, the complete classification of finite simple groups classification of the finite simple groups was achieved, meaning that the building blocks from which all finite groups can be built are now known, as each finite group has a composition series . During the second half of the twentieth century, mathematicians such as Claude Chevalley Chevalley and Robert Steinberg Steinberg also increased our understanding of finite analogs of classical groups , and other related groups. One such family of groups is the family of general linear group s over finite field s. Finite groups often occur when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure ... symmetry , is strongly influenced by the associated Weyl group s. These are finite groups generated by reflections which act on a finite dimensional Euclidean space . The properties of finite ... orders uses the classification of finite simple groups . For any positive integer n there are at most ... of finite simple groups List of finite simple groups Lagrange s theorem group theory Lagrange ... groups Representation theory of finite groups Modular representation theory Monstrous moonshine Profinite group Infinite group theory Finite ring Div col end Notes references External references Number ... for groups of small order Category Finite groups Category Properties of groups ar ca Grup ... more details
uses see Mathematics disambiguation and Math disambiguation . File Euclid.jpg thumb Euclid , Greek ... . ref Mathematics from Greek language Greek m th ma , knowledge, study, learning is the study ... reasoning often provides insight or predictions. Through the use of abstraction mathematics abstraction and logic al reasoning , mathematics developed from counting , calculation , measurement ... mathematics has been a human activity for as far back as History of Mathematics written records exist. Logic Rigorous arguments first appeared in Greek mathematics , most notably in Euclid Euclid s Euclid s Elements Elements . Mathematics developed at a relatively slow pace until the Renaissance , when ... of Mathematics 1. Newton and Leibniz , BBC Radio 4 , 27 09 2010. ref Carl Friedrich Gauss 1777 1855 referred to mathematics as the Queen of the Sciences . ref name Waltershausen Waltershausen ref Benjamin Peirce 1809 1880 called mathematics the science that draws necessary conclusions . ref Peirce, p. 97. ref David Hilbert said of mathematics We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual ... , Basel, Birkh user 1992 . ref Albert Einstein 1879 1955 stated that as far as the laws of mathematics ... . ref name certain Mathematics is used throughout the world as an essential tool in many fields, including natural science , engineering , medicine , and the social sciences . Applied mathematics , the branch of mathematics concerned with application of mathematical knowledge to other fields ... in pure mathematics , or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered. ref Peterson ref Etymology The word mathematics comes from ... of which mean to learn . The word mathematics in Greek came to have the narrower and more technical ... more details
A finite map can be one of the following In computer science , finite map is a synonym for an associative array . A finite map algebraic geometry finite map in algebraic geometry is a surjective regular map algebraic geometry regular map with zero dimensional fibers. disambig ... more details
A finite geometry is any geometry geometric system that has only a finite set finite number of point geometry points . Euclidean geometry , for example, is not finite, because a Euclidean line contains ... . A finite geometry can have any finite number of dimensions. Finite geometries may be constructed via linear algebra , as vector space s over a finite field , and called Galois geometry Galois geometries , or can be defined purely combinatorially. Many, but not all, finite geometries are Galois geometries for example, any finite projective space of dimension three or greater is isomorphic to a projective space over a finite field the projectivization of a vector space over a finite field , so ... planes which are not isomorphic to projective spaces over finite fields, namely the non Desarguesian plane s, so in this case there is a distinction. Finite planes The following remarks apply only to finite planes . There are two kinds of finite plane geometry affine geometry affine and projective ... parallel lines do not exist. Both finite affine plane geometry and finite projective plane geometry ... Finite affine plane of order 2, containing 4 points and 6 lines. Lines of the same color are parallel ..., a finite affine plane of order math n math has math n 2 math points and math n 2 n math lines each ... thumb 200px right Finite affine plane of order 3, containing 9 points and 12 lines. A projective .... This suggests the principle of Duality mathematics Dimension reversing dualities duality for projective ... A finite plane of order n is one such that each line has n points for an affine plane , or such that each line has math n 1 math points for a projective plane . One major open question in finite geometry is Is the order of a finite plane always a prime power? This is conjectured to be true, but has ... over the finite field with math n p k math elements. Planes not derived from finite fields also exist ... as the order of a finite plane. The smallest integer that is not a prime power and not covered ... more details
In abstract algebra , a finite field or Galois field so named in honor of variste Galois is a field mathematics field that contains a finite number of elements. Finite fields are important in number theory ... domain s principal ideal domain s Euclidean domain s field mathematics field s finite fields . Classification The finite fields are classified as follows Harv Jacobson 2009 loc 4.13,p. 287 The order , or number of elements, of a finite field is of the form p sup n sup , where p is a prime number ... algebra Case of fields Prime field . An example of such a finite field is the Ring mathematics ... it generates is the full group mathematics group of automorphisms of the field. Algebraic closure Finite ... mathematics group of every finite field is cyclic, a special case of field mathematics Some first ... arXiv quant ph 9605005 ref The finite fields are classified by size there is exactly one finite field up to isomorphism of size p sup k sup for each prime p and positive integer k . Each finite field ... group of the field is a cyclic group . Wedderburn s little theorem states that the Brauer group of a finite field is trivial, so that every finite division ring is a finite field. Finite fields have applications in many areas of mathematics and computer science, including coding theory , LFSR s, modular representation theory , and the groups of Lie type . Finite fields are an active area of research, including recent results on the Kakeya set Kakeya sets in vector spaces over finite fields Kakeya ... of primitive roots primitive root . Finite fields appear in the following chain of subclass set theory ... prime number p and positive integer n , there exists a finite field with p sup n sup elements. Any two finite fields with the same number of elements are isomorphic . That is, under some renaming ... tables of the other one. This classification justifies using a naming scheme for finite fields that specifies only the order of the field. One notation for a finite field is F sub p ... more details
the finite sum above is replaced by an Series mathematics infinite series . Another way of generalization ...Use mdy dates date September 2011 A finite difference is a mathematical expression of the form f x b &minus f x a . If a finite difference is divided by b &minus a , one gets a difference quotient . The approximation of derivatives by finite differences plays a central role in finite difference method s for the numerical analysis numerical solution of differential ... equations by replacing iteration notation with finite differences. Forward, backward, and central ... this section is linked to further down in the article In an analogous way one can obtain finite ... of finite differences, explained below. If necessary, the finite difference can be centered ... n h fg, x sum limits k 0 n binom n k Delta k h f, x Delta n k h g, x kh . math Finite difference methods An important application of finite differences is in numerical analysis , especially in numerical ... is to replace the derivatives appearing in the differential equation by finite differences that approximate them. The resulting methods are called finite difference method s. Common applications of the finite difference method are in computational science and engineering disciplines, such as thermal ... 0 frac x a h choose k sum j 0 k 1 k j k choose j f a j h . math Calculus of finite differences The forward difference can be considered as a difference Operator mathematics operator , ref George Boole Boole, George , 1872 . A Treatise On The Calculus of Finite Differences , 2nd ed., Macmillan and Company ... calculus of finite differences&ots vOzf92u Ak&sig b vtnZz9U1zQi3hLmq6Rjk6mC8o v onepage Also, Dover ... , and I is the identity operator . The finite difference of higher orders can be defined in recursive ... arsinh tfrac12 delta h . math The calculus of finite differences is related to the umbral calculus ... f x thus map systematically to umbral finite difference analogs involving f xT sub h sub sup 1 ... more details
A finite verb is a verb that is Inflection inflected for grammatical person person and for grammatical tense tense according to the rules and categories of the languages in which it occurs. Finite verbs can form independent clause s, which can stand on their own as complete Sentence linguistics sentence s. The finite forms of a verb are the forms where the verb shows tense, person or number. Non finite verb forms have no person or number, but some types can show tense. Finite verb forms include I go, she goes, he went Non finite verb forms include to go, going, gone Indo European languages In the Indo European language s such as English , only verbs in certain grammatical mood moods are finite. These include the indicative mood expressing a state of affairs e.g., The bulldozer demolished the restaurant, The leaves were yellow and stiff. the Imperative mood imperative mood giving a command e. g., Come here , Be a good boy the subjunctive mood typically used in dependent clauses e. g., It is required ... expressing a wish or hope . Non existent as a mood in English. Verb forms that are non finite verb not finite include the infinitive the participle s e. g., The broken window... , The wheezing gentleman ... complete sentence or clause must contain a finite verb. However, sentences lacking a finite ... where it could have a subject grammar subject and a finite verb form compare I appreciate your help . Finite verbs in syntax Finite verbs play a particularly important role in syntactic analyses of sentence structure. In phrase structure grammar phrase structure grammars , the finite verb is the head of a finite verb phrase VP and as such, it is the head of the entire sentence, and in dependency grammar dependency grammars , the finite verb is the root of the entire clause and is thus the most prominent structural unit in the clause. See also Non finite verb Balancing and deranking Grammatical ... da Finit verbum de Finite Verbform es conjugaci n Formas personales del verbo li Persoensv rm nl ... more details
Finite part may refer to Cauchy principal value Hadamard finite part mathdab Short pages monitor This long comment was added to the page to prevent it from being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Long comment. Please do not remove the monitor template without removing the comment as well. ... more details
In formal language formal language theory , a class of languages math mathcal L math has finite thickness if for every string s , there are only finitely many consistent languages in math mathcal L math . This condition was introduced by Dana Angluin in connection with learning, as a sufficient condition for language identification in the limit . The related notion of M finite thickness We say that math mathcal L math satisfies the MEF condition if for each string s and each consistent language L in the class, there is a minimal consistent language in math mathcal L math , which is a sublanguage of L. Symmetrically, we say that math mathcal L math satisfies the MFF condition if for every string s there are only finitely many minimal consistent languages in math mathcal L math . Finally, math mathcal L math is said to have M finite thickness if it satisfies both the MEF and MFF conditions. Finite thickness implies M finite thickness. However, there are classes that are of M finite thickness but not of finite thickness for example, let math L n math be a class of languages such that math L 0 subseteq L 1 subseteq ldots math . comp sci theory stub Category Formal languages ... more details
orphan date August 2010 refimprove date March 2009 A finite wing is an wing aerodynamic wing with tips that result in trailing vortices . ref http www.aerospaceweb.org question aerodynamics q0167.shtml ref This is in contrast to an infinite wing . References reflist Category Aircraft wing design aviation stub ... more details
Unreferenced date January 2012 In mathematics, a differentiably finite function , also referred to as a D finite function , is a Function mathematics function which represents a solution of a linear differential equation with polynomial coefficients. category Functions and mappings math stub ... more details
Unreferenced date July 2008 In mathematics , a collection math mathcal U math of subsets of a topological space math X math is said to be point finite or a point finite collection if every point of math X math lies in only finitely many members of math mathcal U math . A topological space in which every open cover admits a point finite open refinement topology refinement is called metacompact space 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 space paracompact . Every paracompact space is metacompact. External links PlanetMath attribution id 8398 title point finite Category General topology topology stub ... more details
In the mathematics mathematical field of group theory , a Group mathematics group G is residually finite ... h from G to a finite group, such that math h g neq 1. , math There are a number of equivalent definitions A group is residually finite if for each non identity element in the group, there is a normal subgroup of finite index group theory index not containing that element. A group is residually finite if and only if the intersection of all its subgroups of finite index is trivial. A group is residually finite if and only if the intersection of all its normal subgroups of finite index is trivial. A group is residually finite if and only if it can be embedded inside the direct product of groups direct product of a family of finite groups. Examples Examples of groups that are residually finite are finite group s, free group s, finitely generated group finitely generated nilpotent group s, polycyclic by finite group s, finitely generated group finitely generated General linear group linear groups , and fundamental group s of 3 manifold s. Subgroups of residually finite groups are residually finite, and direct products of residually finite groups are residually finite. Any inverse limit of residually finite groups is residually finite. In particular, all profinite group s are residually finite. Profinite topology Every group G may be made into a topological group by taking as a basis of open neighbourhoods of the identity, the collection of all normal subgroups of finite index in G . The resulting topology is called the profinite topology on G . A group is residually finite ... , for locally extended residually finite . A group in which every conjugacy class is closed in the profinite topology is called conjugacy separable . Varieties of residually finite groups One question ... finite? Two results about these are Any variety comprising only residually finite groups is generated by an A group . For any variety comprising only residually finite groups, it contains a finite ... more details
Orphan date February 2010 In algebra , a ring mathematics ring math R math is said to be stably finite or weakly finite if, for all square matrices A , B of the same size over R , AB 1 implies BA 1. This is a slightly stronger property for a ring than its having the invariant basis number any nontrivial ref A trivial ring is stably finite but doesn t have IBN. ref stably finite ring has IBN. Commutative rings, noetherian ring s and artinian ring s are stably finite. A subring of a stably finite ring and a matrix ring over a stably finite ring is stably finite. A ring satisfying Klein s nilpotence condition is stably finite. Citation needed date February 2010 References Reflist P.M. Cohn 2003 . Basic Algebra, Springer. Category Ring theory Abstract algebra stub ... more details
In mathematics , in the field of group theory , a locally finite group is a type of group mathematics group that can be studied in ways analogous to a finite group . Sylow subgroup s, Carter subgroup s, and abelian subgroup s of locally finite groups have been studied. Definition and first consequences A locally finite group is a group for which every finitely generated group finitely generated subgroup is finite group finite . Since the cyclic subgroup s of a locally finite group are finite, every element has finite order group theory order , and so the group is periodic group periodic . Examples and non examples Examples Every finite group is locally finite Every infinite direct sum of finite groups is locally finite harv Robinson 1996 p 443 Although the direct product may not be. The Pr fer group s are locally finite abelian groups Every Hamiltonian group is locally finite Every periodic solvable group is locally finite harv Dixon 1994 loc Prop. 1.1.5 . Every subgroup of a locally finite group is locally finite. If G is a group and S is a subgroup of G and F is a finite subset of S , the subgroup ... of G contradicting the fact that G is locally finite. Every group has a unique maximal normal locally finite subgroup harv Robinson 1996 p 436 Every periodic group periodic subgroup of the general linear group over the complex numbers is locally finite. Since all locally finite groups are periodic ... Citation title Representation Theory of Finite Groups and Associated Algebras last Curtis first ... No group with an element of infinite order is a locally finite group No nontrivial free group is locally finite A Tarski monster group is periodic, but not locally finite. Properties The class of locally finite groups is closed under subgroups, quotient group quotients , and group extension extensions harv Robinson 1996 p 429 . Locally finite groups satisfy a weaker form of Sylow s theorems . If a locally finite group has a finite p group p subgroup contained in no other p subgroups, then all ... more details
In mathematics , a locally finite measure is a Measure mathematics measure for which every point of the measure space has a Neighbourhood mathematics neighbourhood of Wikt finitefinite measure. Definition Let X , T be a Hausdorff space Hausdorff topological space and let &Sigma be a sigma algebra &sigma algebra on X that contains the topology T so that every open set is a measurable set , and &Sigma is at least as fine as the Borel sigma algebra Borel &sigma algebra on X . A measure signed measure complex measure &mu defined on &Sigma is called locally finite if, for every point p of the space X , there is an open neighbourhood N sub p sub of p such that the &mu measure of N sub p sub is finite. In more condensed notation, &mu is locally finite if and only if math forall p in X, exists N p in T mbox s.t. p in N p mbox and left mu N p right infty. math Examples Any probability measure on X is locally finite, since it assigns unit measure the whole space. Similarly, any measure that assigns finite measure to the whole space is locally finite. Lebesgue measure on Euclidean space is locally finite. By definition, any Radon measure is locally finite. Counting measure is sometimes locally finite and sometimes not counting measure on the integer s with their usual discrete topology is locally finite, but counting measure on the real line with its usual Borel topology is not. See also Inner regular measure Strictly positive measure References Unreferenced date December 2006 Category Measures measure theory fr Mesure localement finie nl Lokaal eindige maat pl Miara lokalnie sko czona ... more details
In the mathematics mathematical field of topology , local finiteness is a property of collections of subset ... 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 mathematics neighbourhood that intersects only finitely many of the sets in the collection. Note that the term Locally finite disambiguation locally finite has different meanings in other mathematical fields. Examples and properties A finite set finite collection of subsets of a topological space is locally finite. Infinite collections can also be locally finite ... n . A countable collection of subsets need not be locally finite, as shown by the collection of all subsets of R of the form &minus n , n with integer n . If a collection of sets is locally finite, the collection of all closures of these sets is also locally finite. The converse, however, can fail if the closures of the sets are not distinct. For example, in the finite complement topology on R the collection of all open sets is not locally finite, but the collection of all closures of these sets is locally finite since the only closures are R and the empty set . Compact spaces No infinite collection of a compact space can be locally finite. Indeed, let G sub a sub be an infinite family of subsets of a space and suppose this collection is locally finite. For each point x of this space ... has a finite subcover, U sub a sub 1 sub sub ...... U sub a sub n sub sub . Since each U sub ... every open cover admits a locally finite open refinement topology refinement is called paracompact space paracompact . Every locally finite collection of subsets of a topological space X is also Point finite collection point finite . A topological space in which every open cover admits a point finite ... cover topology cover of a Lindel f space space can be locally finite, by essentially the same argument ... is locally finite. Closed sets It is clear from the definition of a topology that a finite union of closed ... more details
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