Locally constant function
Encyclopedia
In mathematics
, a function
f from a topological space
A to a set B is called locally constant, if for every a in A there exists a neighborhood U of a, such that f is constant on U.
Every constant function
is locally constant.
Every locally constant function from the real number
s R to R is constant by the connectedness
of R. But the function f from the rationals
Q to R, defined by f(x) = 0 for x < π
, and f(x) = 1 for x > π, is locally constant (here we use the fact that π is irrational
and that therefore the two sets {x∈Q : x < π} and {x∈Q : x > π} are both open
in Q).
If f : A → B is locally constant, then it is constant on any connected component
of A. The converse is true for locally connected spaces (where the connected components are open).
Further examples include the following:
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 axioms hold for this construction, giving us a sheaf of abelian group
s (even commutative ring
s). This sheaf could be written ZX; described by means of stalks we have stalk Zx, a copy of Z at x, for each 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 isn't 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), but from a global point of view exhibit some 'twisting'.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
f from a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
A to a set B is called locally constant, if for every a in A there exists a neighborhood U of a, such that f is constant on U.
Every constant function
Constant function
In mathematics, a constant function is a function whose values do not vary and thus are constant. For example the function f = 4 is constant since f maps any value to 4...
is locally constant.
Every locally constant function from the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s R to R is constant by the connectedness
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
of R. But the function f from the rationals
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
Q to R, defined by f(x) = 0 for x < π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
, and f(x) = 1 for x > π, is locally constant (here we use the fact that π is irrational
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
and that therefore the two sets {x∈Q : x < π} and {x∈Q : x > π} are both open
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
in Q).
If f : A → B is locally constant, then it is constant on any connected component
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
of A. The converse is true for locally connected spaces (where the connected components are open).
Further examples include the following:
- Given a coveringCovering-Mathematics:*In topology:** Covering map, a function from one space to another with uniform local neighborhoods** Cover , a system of sets whose union is a given topological space...
p : C → X, then to each point x of X we can assign the cardinality of the fiber p−1(x) over x; this assignment is locally constant. - A map from a topological space A to a discrete spaceDiscrete spaceIn topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
B is continuous if and only if it is locally constant.
Connection with sheaf theory
There are sheaves of locally constant functions on X. To be more definite, the locally constant integer-valued functions on X form a sheafSheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
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 axioms hold for this construction, giving us a sheaf of abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s (even commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s). This sheaf could be written ZX; described by means of stalks we have stalk Zx, a copy of Z at x, for each 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 isn't constant in this way; but the construction is useful in linking up sheaf cohomology
Sheaf cohomology
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
with homology theory
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
, 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), but from a global point of view exhibit some 'twisting'.