proper map
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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 f : X \to Y 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 Y is compact in X. Other authors call a map f 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 Y 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 Y is locally compact and Hausdorff. Let f : X \to Y be a closed map, such that f^(y) is compact (in X) for all y \in Y. Let K be a compact subset of Y. It remains to show that f^(K) is compact. Let \left\ be an open cover of f^(K). Then for all k \in K this is also an open cover of f^(k). Since the latter is assumed to be compact, it has a finite subcover. In other words, for every k \in K, there exists a finite subset \gamma_k \subseteq A such that f^(k) \subseteq \cup_ U_. The set X \setminus \cup_ U_ is closed in X and its image under f is closed in Y because f is a closed map. Hence the set V_k = Y \setminus f\left(X \setminus \cup_ U_\right) is open in Y. It follows that V_k contains the point k. Now K \subseteq \cup_ V_k and because K is assumed to be compact, there are finitely many points k_1, \dots, k_s such that K \subseteq \cup_^s V_. Furthermore, the set \Gamma = \cup_^s \gamma_ is a finite union of finite sets, which makes \Gamma a finite set. Now it follows that f^(K) \subseteq f^\left( \cup_^s V_ \right) \subseteq \cup_ U_ and we have found a finite subcover of f^(K), which completes the proof. If X is Hausdorff and Y is locally compact Hausdorff then proper is equivalent to . A map is universally closed if for any topological space Z the map f \times \operatorname_Z : X \times Z \to Y \times Z is closed. In the case that Y is Hausdorff, this is equivalent to requiring that for any map Z \to Y the pullback X \times_Y Z \to Z be closed, as follows from the fact that X \times_YZ is a closed subspace of X \times Z. An equivalent, possibly more intuitive definition when X and Y 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 X if, for every compact set S \subseteq X only finitely many points p_i are in S. Then a continuous map f : X \to Y is proper if and only if for every sequence of points \left\ that escapes to infinity in X, the sequence \left\ escapes to infinity in Y.


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 f : X \to Y is called a if for every compact subset K \subseteq Y there exists some compact subset C \subseteq X such that f(C) = K. * A topological space is compact if and only if the map from that space to a single point is proper. * If f : X \to Y is a proper continuous map and Y 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 f 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