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 → S {\displaystyle f:X\to S} be a function from a topological space X {\displaystyle X} into a set S . {\displaystyle S.} If x ∈ X {\displaystyle x\in X} then f {\displaystyle f} is said to be locally constant at x {\displaystyle x} if there exists a neighborhood U ⊆ X {\displaystyle U\subseteq X} of x {\displaystyle x} such that f {\displaystyle f} is constant on U , {\displaystyle U,} which by definition means that f ( u ) = f ( v ) {\displaystyle f(u)=f(v)} for all u , v ∈ U . {\displaystyle u,v\in U.} The function f : X → S {\displaystyle f:X\to S} is called locally constant if it is locally constant at every point x ∈ X {\displaystyle 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 {\displaystyle \mathbb {R} } to R {\displaystyle \mathbb {R} } is constant, by the connectedness of R . {\displaystyle \mathbb {R} .} But the function f : Q → R {\displaystyle f:\mathbb {Q} \to \mathbb {R} } from the rationals Q {\displaystyle \mathbb {Q} } to R , {\displaystyle \mathbb {R} ,} defined by f ( x ) = 0 for x < π , {\displaystyle f(x)=0{\text{ for }}x<\pi ,} and f ( x ) = 1 for x > π , {\displaystyle f(x)=1{\text{ for }}x>\pi ,} is locally constant (this uses the fact that π {\displaystyle \pi } is irrational and that therefore the two sets { x ∈ Q : x < π } {\displaystyle \{x\in \mathbb {Q} :x<\pi \}} and { x ∈ Q : x > π } {\displaystyle \{x\in \mathbb {Q} :x>\pi \}} are both open in Q {\displaystyle \mathbb {Q} } ).
If f : A → B {\displaystyle f:A\to B} is locally constant, then it is constant on any connected component of A . {\displaystyle 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 → X , {\displaystyle p:C\to X,} then to each point x ∈ X {\displaystyle x\in X} we can assign the cardinality of the fiber p − 1 ( x ) {\displaystyle p^{-1}(x)} over x {\displaystyle x} ; this assignment is locally constant.
- A map from a topological space A {\displaystyle A} to a discrete space B {\displaystyle B} is continuous if and only if it is locally constant.
Connection with sheaf theory
There are sheaves of locally constant functions on X . {\displaystyle X.} To be more definite, the locally constant integer-valued functions on X {\displaystyle X} form a sheaf in the sense that for each open set U {\displaystyle U} of X {\displaystyle X} 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 Z X {\displaystyle Z_{X}} ; described by means of stalks we have stalk Z x , {\displaystyle Z_{x},} a copy of Z {\displaystyle Z} at x , {\displaystyle x,} for each x ∈ X . {\displaystyle x\in X.} 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 x {\displaystyle x} ), but from a global point of view exhibit some 'twisting'.
See also
- Liouville's theorem (complex analysis) – Theorem in complex analysis
- Locally constant sheaf
References
Hartshorne, Robin (1977). Algebraic Geometry. Springer. p. 62. ↩