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 between topological spaces is called proper if inverse images of 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".

Some authors call a function $f : X \to Y$ between two topological spaces {{em|proper}} if the preimage of every compact set in $Y$ is compact in $X.$

Other authors call a map $f$ {{em|proper}} if it is continuous and {{em|closed with compact fibers}}; that is if it is a continuous closed map and the preimage of every point in $Y$ is compact. The two definitions are equivalent if $Y$ is locally compact and Hausdorff.

Let $f : X \to Y$ be a closed map, such that $f^{-1}(y)$ is compact (in $X$ ) for all $y \in Y.$ Let $K$ be a compact subset of $Y.$ It remains to show that $f^{-1}(K)$ is compact.

Let $\left\{U_a : a \in A\right\}$ be an open cover of $f^{-1}(K).$ Then for all $k \in K$ this is also an open cover of $f^{-1}(k).$ 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 $f^{-1}(k) \subseteq \cup_{a \in \gamma_k} U_{a}.$

The set $X \setminus \cup_{a \in \gamma_k} U_{a}$ is closed in $X$ and its image under $f$ is closed in $Y$ because $f$ is a closed map. Hence the set

$V_k = Y \setminus f\left(X \setminus \cup_{a \in \gamma_k} U_{a}\right)$

is open in $Y.$ It follows that $V_k$ contains the point $k.$

Now $K \subseteq \cup_{k \in K} V_k$ and because $K$ is assumed to be compact, there are finitely many points $k_1, \dots, k_s$ such that $K \subseteq \cup_{i =1}^s V_{k_i}.$ Furthermore, the set $\Gamma = \cup_{i=1}^s \gamma_{k_i}$ is a finite union of finite sets, which makes $\Gamma$ a finite set.

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

If $X$ is Hausdorff and $Y$ is locally compact Hausdorff then proper is equivalent to {{em|universally closed}}. A map is universally closed if for any topological space $Z$ the map $f \times \operatorname{id}_Z : X \times Z \to Y \times Z$ 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 spaces is as follows: we say an infinite sequence of points $\{p_i\}$ in a topological space $X$ {{em|escapes to infinity}} 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 $\left\{p_i\right\}$ that escapes to infinity in $X,$ the sequence $\left\{f\left(p_i\right)\right\}$ escapes to infinity in $Y.$

Properties

Every continuous map from a compact space to a Hausdorff space is both proper and closed.
Every surjective proper map is a compact covering map.
A map $f : X \to Y$ is called a {{em|compact covering}} 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 (this includes Hausdorff spaces that are either first-countable or locally compact), then $f$ is closed.{{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 }}

Generalization

It is possible to generalize

the notion of proper maps of topological spaces to locales and topoi, see {{Harv|Johnstone|2002}}.

Citations

References

{{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. 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}}

Category:Theory of continuous functions