Locally constant function

In mathematics, a locally constant function is a function from a topological space into a set with the property that around every point of its domain, there exists some neighborhood of that point on which it restricts to a constant function.

The signum function restricted to the domain is locally constant.

Definition

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Let   be a function from a topological space   into a set   If   then   is said to be locally constant at   if there exists a neighborhood   of   such that   is constant on   which by definition means that   for all   The function   is called locally constant if it is locally constant at every point   in its domain.

Examples

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Every constant function is locally constant. The converse will hold if its domain is a connected space.

Every locally constant function from the real numbers   to   is constant, by the connectedness of   But the function   from the rationals   to   defined by   and   is locally constant (this uses the fact that   is irrational and that therefore the two sets   and   are both open in  ).

If   is locally constant, then it is constant on any connected component of   The converse is true for locally connected spaces, which are spaces whose connected components are open subsets.

Further examples include the following:

  • Given a covering map   then to each point   we can assign the cardinality of the fiber   over  ; this assignment is locally constant.
  • A map from a topological space   to a discrete space   is continuous if and only if it is locally constant.

Connection with sheaf theory

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There are sheaves of locally constant functions on   To be more definite, the locally constant integer-valued functions on   form a sheaf in the sense that for each open set   of   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 groups (even commutative rings).[1] This sheaf could be written  ; described by means of stalks we have stalk   a copy of   at   for each   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 is not 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  ), but from a global point of view exhibit some 'twisting'.

See also

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References

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  1. ^ Hartshorne, Robin (1977). Algebraic Geometry. Springer. p. 62.