Template:Short description Template:More citations needed Template:Redirect
.svg){kind=link}
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.
DefinitionEdit
Let <math>f : X \to S</math> be a function from a topological space <math>X</math> into a set <math>S.</math> If <math>x \in X</math> then <math>f</math> is said to be locally constant at <math>x</math> if there exists a neighborhood <math>U \subseteq X</math> of <math>x</math> such that <math>f</math> is constant on <math>U,</math> which by definition means that <math>f(u) = f(v)</math> for all <math>u, v \in U.</math> The function <math>f : X \to S</math> is called locally constant if it is locally constant at every point <math>x \in X</math> in its domain.
ExamplesEdit
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 <math>\R</math> to <math>\R</math> is constant, by the connectedness of <math>\R.</math> But the function <math>f : \Q \to \R</math> from the rationals <math>\Q</math> to <math>\R,</math> defined by <math>f(x) = 0 \text{ for } x < \pi,</math> and <math>f(x) = 1 \text{ for } x > \pi,</math> is locally constant (this uses the fact that <math>\pi</math> is irrational and that therefore the two sets <math>\{ x \in \Q : x < \pi \}</math> and <math>\{ x \in \Q : x > \pi \}</math> are both open in <math>\Q</math>).
If <math>f : A \to B</math> is locally constant, then it is constant on any connected component of <math>A.</math> 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 <math>p : C \to X,</math> then to each point <math>x \in X</math> we can assign the cardinality of the fiber <math>p^{-1}(x)</math> over <math>x</math>; this assignment is locally constant.
- A map from a topological space <math>A</math> to a discrete space <math>B</math> is continuous if and only if it is locally constant.
Connection with sheaf theoryEdit
There are Template:Em of locally constant functions on <math>X.</math> To be more definite, the locally constant integer-valued functions on <math>X</math> form a sheaf in the sense that for each open set <math>U</math> of <math>X</math> we can form the functions of this kind; and then verify that the sheaf Template:Em hold for this construction, giving us a sheaf of abelian groups (even commutative rings).<ref>Template:Cite book</ref> This sheaf could be written <math>Z_X</math>; described by means of Template:Em we have stalk <math>Z_x,</math> a copy of <math>Z</math> at <math>x,</math> for each <math>x \in X.</math> This can be referred to a Template:Em, meaning exactly Template:Em 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 Template:Em look like such 'harmless' sheaves (near any <math>x</math>), but from a global point of view exhibit some 'twisting'.