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local property wikipedia 76 captures 13 sep 2006 20 jul 2026 aug sep oct 04 2021 2022 2023 success fail about this capture timestamps the wayback machine http web archive org web 20220904144237 https en wikipedia org wiki local_property local property from wikipedia the free encyclopedia jump to navigation jump to search in mathematics a mathematical object is said to satisfy a property locally if the property is satisfied on some limited immediate portions of the object e g on some sufficiently small or arbitrarily small neighborhoods of points contents 1 properties of a point on a function 2 properties of a single space 2 1 examples 3 properties of a pair of spaces 4 properties of infinite groups 5 properties of finite groups 6 properties of commutative rings 7 see also 8 references properties of a point on a function edit perhaps the best known example of the idea of locality lies in the concept of local minimum or local maximum which is a point in a function whose functional value is the smallest resp largest within an immediate neighborhood of points 1 this is to be contrasted with the idea of global minimum or global maximum which corresponds to the minimum resp maximum of the function across its entire domain 2 3 properties of a single space edit a topological space is sometimes said to exhibit a property locally if the property is exhibited near each point in one of the following ways each point has a neighborhood exhibiting the property each point has a neighborhood base of sets exhibiting the property here note that condition 2 is for the most part stronger than condition 1 and that extra caution should be taken to distinguish between the two for example some variation in the definition of locally compact can arise as a result of the different choices of these conditions examples edit locally compact topological spaces 4 locally connected and locally path connected topological spaces locally hausdorff locally regular locally normal etc locally metrizable properties of a pair of spaces edit given some notion of equivalence e g homeomorphism diffeomorphism isometry between topological spaces two spaces are said to be locally equivalent if every point of the first space has a neighborhood which is equivalent to a neighborhood of the second space for instance the circle and the line are very different objects one cannot stretch the circle to look like the line nor compress the line to fit on the circle without gaps or overlaps however a small piece of the circle can be stretched and flattened out to look like a small piece of the line for this reason one may say that the circle and the line are locally equivalent similarly the sphere and the plane are locally equivalent a small enough observer standing on the surface of a sphere e g a person and the earth would find it indistinguishable from a plane properties of infinite groups edit for an infinite group a small neighborhood is taken to be a finitely generated subgroup an infinite group is said to be locally p if every finitely generated subgroup is p for instance a group is locally finite if every finitely generated subgroup is finite and a group is locally soluble if every finitely generated subgroup is soluble properties of finite groups edit for finite groups a small neighborhood is taken to be a subgroup defined in terms of a prime number p usually the local subgroups the normalizers of the nontrivial p subgroups in which case a property is said to be local if it can be detected from the local subgroups global and local properties formed a significant portion of the early work on the classification of finite simple groups which was carried out during the 1960s properties of commutative rings edit main article local ring for commutative rings ideas of algebraic geometry make it natural to take a small neighborhood of a ring to be the localization at a prime ideal in which case a property is said to be local if it can be detected from the local rings for instance being a flat module over a commutative ring is a local property but being a free module is not for more see localization of a module see also edit local path connectedness references edit definition of local maximum dictionary com www dictionary com retrieved 2019 11 30 weisstein eric w local minimum mathworld wolfram com retrieved 2019 11 30 maxima minima and saddle points khan academy retrieved 2019 11 30 weisstein eric w locally compact mathworld wolfram com retrieved 2019 11 30 retrieved from https en wikipedia org w index php title local_property oldid 1066449263 categories general topology homeomorphisms navigation menu personal tools not logged in talk contributions create account log in namespaces article talk english views read edit view history more search navigation main page contents current events random article about wikipedia contact us donate contribute help learn to edit community portal recent changes upload file tools what links here related changes upload file special pages permanent link page information cite this page wikidata item print export download as pdf printable version languages deutsch español français polski português 文言 中文 edit links this page was last edited on 18 january 2022 at 12 07 utc text is available under the creative commons attribution sharealike license 3 0 additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia mobile view developers statistics cookie statement