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.

Definition

Let $f : X \to S$ be a function from a topological space $X$ into a set $S.$
If $x \in X$ then $f$ is said to locally constant at $x$ if there exists a neighborhood $U \subseteq X$ of $x$ such that $f$ is constant on $U,$ which by definition means that $f(u) = f(v)$ for all $u, v \in U.$
The function $f : X \to S$ is called locally constant if it is locally constant at every point $x \in X$ 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 $\R$ to $\R$ is constant, by the connectedness of $\R.$ But the function $f : \Q \to \R$ from the rationals $\Q$ to $\R,$ defined by $f(x) = 0 \text x < \pi,$ and $f(x) = 1 \text x > \pi,$ is locally constant (this uses the fact that $\pi$ is irrational and that therefore the two sets $\$ and $\$ are both open in $\Q$).

If $f : A \to B$ is locally constant, then it is constant on any connected component of $A.$ 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 $p : C \to X,$ then to each point $x \in X$ we can assign the cardinality of the fiber $p^(x)$ over $x$; this assignment is locally constant.
* A map from a topological space $A$ to a discrete space $B$ is continuous if and only if it is locally constant.

Connection with sheaf theory

There are 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 hold for this construction, giving us a sheaf of abelian groups (even commutative rings). This sheaf could be written $Z_X$; described by means of we have stalk $Z_x,$ a copy of $Z$ at $x,$ for each $x \in X.$ 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 look like such 'harmless' sheaves (near any $x$), but from a global point of view exhibit some 'twisting'.

See also

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{{DEFAULTSORT:Locally Constant Function
Sheaf theory