In
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, Lebesgue's number lemma, named after
Henri Lebesgue
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
, is a useful tool in the study of
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s. It states:
:If the metric space
is compact and an
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ...
of
is given, then there exists a number
such that every
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of
having
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...
less than
is contained in some member of the cover.
Such a number
is called a Lebesgue number of this cover. The notion of a Lebesgue number itself is useful in other applications as well.
Proof
Let
be an open cover of
. Since
is compact we can extract a finite subcover
.
If any one of the
's equals
then any
will serve as a Lebesgue number.
Otherwise for each
, let
, note that
is not empty, and define a function
by
:
Since
is continuous on a compact set, it attains a minimum
.
The key observation is that, since every
is contained in some
, the
extreme value theorem
In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval ,b/math>, then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in ,b/math> suc ...
shows
. Now we can verify that this
is the desired Lebesgue number.
If
is a subset of
of diameter less than
, then there exists
such that
, where
denotes the ball of radius
centered at
(namely, one can choose
as any point in
). Since
there must exist at least one
such that
. But this means that
and so, in particular,
.
References
*
Theorems in topology
Lemmas
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