View source for Proper map

{{short description|Map between topological spaces with the property that the preimage of every compact is compact}}
{{About|the concept in [[topology]]|the concept in [[convex analysis]]|proper convex function}}

In [[mathematics]], a [[function (mathematics)|function]] between [[topological space]]s is called '''proper''' if [[inverse image]]s of [[compact space|compact subsets]] are compact.{{sfn|Lee|2012|p=610|loc=above Prop. A.53}} In [[algebraic geometry]], the [[analogous]] concept is called a [[proper morphism]].

==Definition==

There are several competing definitions of a "proper [[Function (mathematics)|function]]". 
Some authors call a function <math>f : X \to Y</math> between two [[topological space]]s '''{{em|proper}}''' if the [[preimage]] of every [[Compact space|compact]] set in <math>Y</math> is compact in <math>X.</math>
Other authors call a map <math>f</math> {{em|proper}} if it is continuous and '''{{em|closed with compact fibers}}'''; that is if it is a [[Continuous map|continuous]] [[closed map]] and the preimage of every point in <math>Y</math> is [[Compact set|compact]]. The two definitions are equivalent if <math>Y</math> is [[Locally compact space|locally compact]] and [[Hausdorff space|Hausdorff]].
{{Collapse top|title=Partial proof of equivalence}}
Let <math>f : X \to Y</math> be a closed map, such that <math>f^{-1}(y)</math> is compact (in <math>X</math>) for all <math>y \in Y.</math> Let <math>K</math> be a compact subset of <math>Y.</math> It remains to show that <math>f^{-1}(K)</math> is compact.

Let <math>\left\{U_a : a \in A\right\}</math> be an open cover of <math>f^{-1}(K).</math> Then for all <math>k \in K</math> this is also an open cover of <math>f^{-1}(k).</math> Since the latter is assumed to be compact, it has a finite subcover. In other words, for every <math>k \in K,</math> there exists a finite subset <math>\gamma_k \subseteq A</math> such that <math>f^{-1}(k) \subseteq \cup_{a \in \gamma_k} U_{a}.</math>
The set <math>X \setminus \cup_{a \in \gamma_k} U_{a}</math> is closed in <math>X</math> and its image under <math>f</math> is closed in <math>Y</math> because <math>f</math> is a closed map. Hence the set
<math display=block>V_k = Y \setminus f\left(X \setminus \cup_{a \in \gamma_k} U_{a}\right)</math> 
is open in <math>Y.</math> It follows that <math>V_k</math> contains the point <math>k.</math>
Now <math>K \subseteq \cup_{k \in K} V_k</math> and because <math>K</math> is assumed to be compact, there are finitely many points <math>k_1, \dots, k_s</math> such that <math>K \subseteq \cup_{i =1}^s V_{k_i}.</math> Furthermore, the set <math>\Gamma = \cup_{i=1}^s \gamma_{k_i}</math> is a finite union of finite sets, which makes <math>\Gamma</math> a finite set.

Now it follows that <math>f^{-1}(K) \subseteq f^{-1}\left( \cup_{i=1}^s V_{k_i} \right) \subseteq \cup_{a \in \Gamma} U_{a}</math> and we have found a finite subcover of <math>f^{-1}(K),</math> which completes the proof.
{{Collapse bottom}}

If <math>X</math> is Hausdorff and <math>Y</math> is locally compact Hausdorff then proper is equivalent to '''{{em|universally closed}}'''. A map is universally closed if for any topological space <math>Z</math> the map <math>f \times \operatorname{id}_Z : X \times Z \to Y \times Z</math> is closed. In the case that <math>Y</math> is Hausdorff, this is equivalent to requiring that for any map <math>Z \to Y</math> the pullback <math>X \times_Y Z \to Z</math> be closed, as follows from the fact that <math>X \times_YZ</math> is a closed subspace of <math>X \times Z.</math>

An equivalent, possibly more intuitive definition when <math>X</math> and <math>Y</math> are [[metric space]]s is as follows: we say an infinite sequence of points <math>\{p_i\}</math> in a topological space <math>X</math> '''{{em|escapes to infinity}}''' if, for every compact set <math>S \subseteq X</math> only finitely many points <math>p_i</math> are in <math>S.</math> Then a continuous map <math>f : X \to Y</math> is proper if and only if for every sequence of points <math>\left\{p_i\right\}</math> that escapes to infinity in <math>X,</math> the sequence <math>\left\{f\left(p_i\right)\right\}</math> escapes to infinity in <math>Y.</math>

==Properties==

* Every continuous map from a compact space to a [[Hausdorff space]] is both proper and [[Closed map|closed]].
* Every [[Surjective map|surjective]] proper map is a compact covering map.
** A map <math>f : X \to Y</math> is called a '''{{em|compact covering}}''' if for every compact subset <math>K \subseteq Y</math> there exists some compact subset <math>C \subseteq X</math> such that <math>f(C) = K.</math> 
* A topological space is compact if and only if the map from that space to a single point is proper.
* If <math>f : X \to Y</math> is a proper continuous map and <math>Y</math> is a [[compactly generated Hausdorff space]] (this includes Hausdorff spaces that are either [[First-countable space|first-countable]] or [[Locally compact space|locally compact]]), then <math>f</math> is closed.<ref name=palais>{{cite journal|last=Palais|first=Richard S.|author-link=Richard Palais| title=When proper maps are closed|journal=[[Proceedings of the American Mathematical Society]]|year=1970|volume=24|issue=4|pages=835–836|doi=10.1090/s0002-9939-1970-0254818-x|doi-access=free|mr=0254818 }}</ref>

==Generalization==

It is possible to generalize 
the notion of proper maps of topological spaces to [[Pointless topology|locales]] and [[topos|topoi]], see {{Harv|Johnstone|2002}}.

==See also==

* {{annotated link|Almost open map}}
* {{annotated link|Open and closed maps}}
* {{annotated link|Perfect map}}
* {{annotated link|Topology glossary}}

==Citations==

{{reflist}}

==References==
{{sfn whitelist |CITEREFLee2012}}
{{refbegin}}
* {{cite book | last1=Bourbaki | first1=Nicolas | author1-link = Nicolas Bourbaki | title=General topology. Chapters 5–10 | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Elements of Mathematics | isbn=978-3-540-64563-4 |mr=1726872 | year=1998}}
* {{cite book |last=Johnstone |first=Peter |author-link=Peter Johnstone (mathematician)| title=Sketches of an elephant: a topos theory compendium |publisher=[[Oxford University Press]] |location=Oxford |year=2002 |isbn=0-19-851598-7 }}, esp. section C3.2 "Proper maps"
* {{cite book |last=Brown |first=Ronald |author-link=Ronald Brown (mathematician)| title=Topology and groupoids |publisher=[[Booksurge]] |location= North Carolina |year=2006 |isbn=1-4196-2722-8 }}, esp. p.&nbsp;90 "Proper maps" and the Exercises to Section 3.6.
* {{cite journal |last=Brown |first=Ronald |author-link=Ronald Brown (mathematician)|  title=Sequentially proper maps and a sequential compactification| journal= [[Journal of the London Mathematical Society]] | series=Second series|volume=7 | issue=3 |year=1973|pages= 515-522 | doi=10.1112/jlms/s2-7.3.515}}
* {{Lee Introduction to Smooth Manifolds|second}}
{{refend}}

{{Topology}}

[[Category:Theory of continuous functions]]