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[[de:Lokal konstante Funktion]]▼
In [[mathematics]], a [[function (mathematics)|function]] ''f'' from a [[topological space]] ''A'' to a [[set]] ''B'' is called '''locally constant''', [[iff]] for every ''a'' in ''A'' there exists a [[neighborhood (topology)|neighborhood]] ''U'' of ''a'', such that ''f'' is constant on ''U''.
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There are ''sheaves'' of locally constant functions on ''X''. To be more definite, the locally constant integer-valued functions on ''X'' form a [[sheaf]] in the sense that for each open set ''U'' of ''X'' we can form the functions of this kind; and then verify that the sheaf ''axioms'' hold for this construction, giving us a sheaf of [[abelian group]]s (even [[commutative ring]]s). This sheaf could be written ''Z''<sub>''X''</sub>; described by means of ''stalks'' we have stalk ''Z''<sub>''x''</sub>, a copy of ''Z'' at ''x'', for each ''x'' in ''X''. This can be referred to a ''constant sheaf'', meaning exactly ''sheaf of locally constant functions'' taking their values in the (same) group. The typical sheaf of course isn't constant in this way; but the construction is useful in linking up [[sheaf cohomology]] with [[homology theory]], and in logical applications of sheaves. The idea of [[local coefficient system]] is that we can have a theory of sheaves that ''locally'' look like such 'harmless' sheaves (near any ''x''), but from a global point of view exhibit some 'twisting'.
[[Category:Sheaf theory]]
▲[[de:Lokal konstante Funktion]]
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