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SirJective (talk | contribs) m +de:, if constant then locally constant |
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* Given a [[covering]] ''p'' : ''C'' → ''X'', then to each point ''x'' of ''X'' we can assign the [[cardinality]] of the [[fibre]] ''p''<sup>-1</sup>(''x'') over ''x''; this assignment is locally constant.
*A map from the topological space ''A'' to a [[discrete space]] ''B'' is [[continuous]] if and only if it is locally constant.
==Connection with sheaf theory==
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''</sup>, 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'.
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