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Encyclopedia > Locally

In mathematics, something is said to occur locally in the category of topological spaces if it occurs on "small enough" open sets. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...

Given some notion of equivalence (e.g., homeomorphism, diffeomorphism, isometry) between topological spaces, two spaces are locally equivalent if every point of the first space has a neighborhood which is equivalent to a neighborhood of the second space. This word should not be confused with homomorphism. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

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 are 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.

## Unary relation

If P Is a property of topological spaces, then a space is sometimes said to be "locally P" if every point of the space has a neighborhood system of sets with property P. Such is the case with In topology and related areas of mathematics, the neighbourhood system or neighbourhood filter for a point x is the collection of all neighbourhoods for the point x. ...

An exception is a locally closed subset of a topological space, which is simply the intersection of an open set and a closed set. In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... This is a glossary of some terms used in the branch of mathematics known as topology. ...

Results from FactBites:

 localization - definition of localization in Encyclopedia (229 words) Often, localization is the adaption of an object to a locality. In telecommunications, localization is a technique for determining the location of a user of a GSM cell phone. In mathematics, localization is a certain technique in abstract algebra; see also localization of a module, localization of a category.
 Taiwanese localization movement - encyclopedia article about Taiwanese localization movement. (1899 words) Localization (本土化; POJ: pún-thó·-hòa; Pinyin: Běntǔ huà) is a political term used by advocates of Taiwan independence to support their view of Taiwan as not part of China. Localization rejects a Han Chinese identity in favor of a monolithic officially sponsored "Holo" identity, which its supporters equate with a "Taiwanese" identity. The roots of the localization movement began during the Japanese era in Taiwan 1895 to 1945, when groups organized to lobby the imperial government for greater Taiwanese autonomy and home rule.
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