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 ...
, a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
between
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s is called proper if
inverse image
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
s of
compact subsets are compact. In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, the
analogous
Analogy (from Greek ''analogia'', "proportion", from ''ana-'' "upon, according to" lso "against", "anew"+ ''logos'' "ratio" lso "word, speech, reckoning" is a cognitive process of transferring information or meaning from a particular subject ...
concept is called a
proper morphism In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces.
Some authors call a proper variety over a field ''k'' a complete variety. For example, every projective variety over a field ...
.
Definition
There are several competing definitions of a "proper
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
".
Some authors call a function
between two
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s if the
preimage
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
of every
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 ...
set in
is compact in
Other authors call a map
if it is continuous and ; that is if it is a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
closed map
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y.
Likewise, ...
and the preimage of every point in
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 ...
. The two definitions are equivalent if
is
locally compact and
Hausdorff.
Let
be a closed map, such that
is compact (in
) for all
Let
be a compact subset of
It remains to show that
is compact.
Let
be an open cover of
Then for all
this is also an open cover of
Since the latter is assumed to be compact, it has a finite subcover. In other words, for every
there exists a finite subset
such that
The set
is closed in
and its image under
is closed in
because
is a closed map. Hence the set
is open in
It follows that
contains the point
Now
and because
is assumed to be compact, there are finitely many points
such that
Furthermore, the set
is a finite union of finite sets, which makes
a finite set.
Now it follows that
and we have found a finite subcover of
which completes the proof.
If
is Hausdorff and
is locally compact Hausdorff then proper is equivalent to . A map is universally closed if for any topological space
the map
is closed. In the case that
is Hausdorff, this is equivalent to requiring that for any map
the pullback
be closed, as follows from the fact that
is a closed subspace of
An equivalent, possibly more intuitive definition when
and
are
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 is as follows: we say an infinite sequence of points
in a topological space
if, for every compact set
only finitely many points
are in
Then a continuous map
is proper if and only if for every sequence of points
that escapes to infinity in
the sequence
escapes to infinity in
Properties
* Every continuous map from a compact space to a
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
is both proper and
closed.
* Every
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
proper map is a compact covering map.
** A map
is called a if for every compact subset
there exists some compact subset
such that
* A topological space is compact if and only if the map from that space to a single point is proper.
* If
is a proper continuous map and
is a
compactly generated Hausdorff space
In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space ''X'' is compactly generated if it satisfies the following condition:
:A subsp ...
(this includes Hausdorff spaces that are either
first-countable
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...
or
locally compact), then
is closed.
Generalization
It is possible to generalize
the notion of proper maps of topological spaces to
locales and
topoi
In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a noti ...
, see .
See also
*
*
*
*
Citations
References
*
* , esp. section C3.2 "Proper maps"
* , esp. p. 90 "Proper maps" and the Exercises to Section 3.6.
*
*
{{Topology
Theory of continuous functions