Concept
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. Let be a function from a topological space into a set If then is said to 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. 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. There are 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 hold for this construction, giving us a sheaf of abelian groups (even commutative rings). This sheaf could be written ; described by means of we have stalk a copy of at for each 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.
