Locally constant function explained
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
be a function from a
topological space
into a
set
If
then
is said to be
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.
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
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:
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.
Connection with sheaf theory
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).
[1] 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. The idea of
local coefficient system is that we can have a theory of sheaves that look like such 'harmless' sheaves (near any
), but from a global point of view exhibit some 'twisting'.
See also
Notes and References
- Book: Hartshorne . Robin . Algebraic Geometry . 1977 . Springer . 62.
