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.

Definition

be a function from a topological space into a set is said to be if there exists a neighborhood is called locally constant in its domain.

Examples

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 is constant, by the connectedness from the rationals is irrational are both open).

is locally constant, then it is constant on any connected component 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 we can assign the cardinality of the fiber; this assignment is locally constant.

• to a discrete space is continuous if and only if it is locally constant.

Connection with sheaf theory

There are form a sheaf we can form the functions of this kind; and then verify that the sheaf hold for this construction, giving us a sheaf of abelian groups (even commutative rings).; described by means of This can be referred to a , meaning exactly 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 ), but from a global point of view exhibit some 'twisting'.

See also

• Locally constant sheaf

References