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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
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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
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of points. 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.
258:. 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
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if every finitely generated subgroup is finite, and a group is locally soluble if every finitely generated subgroup is
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of a sphere (e.g., a person and the Earth) would find it indistinguishable from a plane.
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and the plane are locally equivalent. A small enough observer standing on the
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Perhaps the best-known example of the idea of locality lies in the concept of
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can arise as a result of the different choices of these conditions.
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make it natural to take a "small neighborhood" of a ring to be the
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may say that the circle and the line are locally equivalent.
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over a commutative ring is a local property, but being a
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338:"Definition of local-maximum | Dictionary.com"
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262:, which was carried out during the 1960s.
198:, a "small neighborhood" is taken to be a
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151:Given some notion of equivalence (e.g.,
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260:classification of finite simple groups
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387:"Maxima, minima, and saddle points"
205:. An infinite group is said to be
50:Properties of a point on a function
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276:For commutative rings, ideas of
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266:Properties of commutative rings
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147:Properties of a pair of spaces
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190:Properties of infinite groups
216:. For instance, a group is
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228:Properties of finite groups
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292:. For instance, being a
314:Local path connectedness
302:Localization of a module
90:exhibiting the property;
319:Local-global principle
300:is not. For more, see
125:Locally path-connected
367:mathworld.wolfram.com
361:Weisstein, Eric W.
133:, Locally regular,
342:www.dictionary.com
278:algebraic geometry
251:of the nontrivial
200:finitely generated
170:For instance, the
165:topological spaces
141:Locally metrizable
127:topological spaces
117:topological spaces
37:sufficiently small
131:Locally Hausdorff
121:Locally connected
95:neighborhood base
93:Each point has a
86:Each point has a
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394:. Retrieved
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46:of points).
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298:free module
294:flat module
290:local rings
286:prime ideal
249:normalizers
29:mathematics
410:Categories
396:2019-11-30
372:2019-11-30
347:2019-11-30
325:References
272:local ring
256:-subgroups
163:) between
308:See also
207:locally
203:subgroup
161:isometry
109:Examples
222:soluble
194:For an
184:surface
80:locally
33:locally
18:Locally
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