Proper map

In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. 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 if the preimage of every compact 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 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 $f^(y)$ 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 spaces 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 is both proper and closed.
* Every surjective 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 (this includes Hausdorff spaces that are either first-countable 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, see .

See also

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Citations

References

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* , esp. section C3.2 "Proper maps"
* , esp. p. 90 "Proper maps" and the Exercises to Section 3.6.
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{{Topology

Theory of continuous functions