In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Heine–Cantor theorem, named after
Eduard Heine
Heinrich Eduard Heine (16 March 1821 – 21 October 1881) was a German mathematician.
Heine became known for results on special functions and in real analysis. In particular, he authored an important treatise on spherical harmonics and Legen ...
and
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
, states that if
is a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
between two
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
and
, and
is
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 ...
, then
is
uniformly continuous
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
. An important special case is that every continuous function from a
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
interval to the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s is uniformly continuous.
Proof
Suppose that
and
are two
metric spaces
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 ...
with metrics
and
, respectively. Suppose further that a function
is continuous and
is compact. We want to show that
is
uniformly continuous
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
, that is, for every positive real number
there exists a positive real number
such that for all points
in the
function domain
In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by \operatorname(f) or \operatornamef, where is the function.
More precisely, given a function f\colon X\to Y, the domain of is . ...
,
implies that
.
Consider some positive real number
. By
continuity, for any point
in the domain
, there exists some positive real number
such that
when
, i.e., a fact that
is within
of
implies that
is within
of
.
Let
be the
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' (YF ...
-neighborhood of
, i.e. the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
:
Since each point
is contained in its own
, we find that the collection
is an open
cover
Cover or covers may refer to:
Packaging
* Another name for a lid
* Cover (philately), generic term for envelope or package
* Album cover, the front of the packaging
* Book cover or magazine cover
** Book design
** Back cover copy, part of co ...
of
. Since
is compact, this cover has a finite subcover
where
. Each of these open sets has an associated radius
. Let us now define
, i.e. the minimum radius of these open sets. Since we have a finite number of positive radii, this minimum
is well-defined and positive. We now show that this
works for the definition of uniform continuity.
Suppose that
for any two
in
. Since the sets
form an open (sub)cover of our space
, we know that
must lie within one of them, say
. Then we have that
. The
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
then implies that
:
implying that
and
are both at most
away from
. By definition of
, this implies that
and
are both less than
. Applying the triangle inequality then yields the desired
:
For an alternative proof in the case of