Proper map

In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact.Template:Sfn 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 <math>f : X \to Y</math> between two topological spaces Template:Em if the preimage of every compact set in <math>Y</math> is compact in <math>X.</math> Other authors call a map <math>f</math> Template:Em if it is continuous and Template:Em; that is if it is a continuous closed map and the preimage of every point in <math>Y</math> is compact. The two definitions are equivalent if <math>Y</math> is locally compact and Hausdorff. Template:Collapse top 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. Template:Collapse bottom

If <math>X</math> is Hausdorff and <math>Y</math> is locally compact Hausdorff then proper is equivalent to Template:Em. 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 spaces is as follows: we say an infinite sequence of points <math>\{p_i\}</math> in a topological space <math>X</math> Template:Em 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.
• Every surjective proper map is a compact covering map.
  • A map <math>f : X \to Y</math> is called a Template:Em 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 or locally compact), then <math>f</math> is closed.<ref name=palais>Template:Cite journal</ref>

Generalization

It is possible to generalize the notion of proper maps of topological spaces to locales and topoi, see Template:Harv.

See also

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References

• Template:Cite book
• Template:Cite book, esp. section C3.2 "Proper maps"
• Template:Cite book, esp. p. 90 "Proper maps" and the Exercises to Section 3.6.
• Template:Cite journal
• Template:Lee Introduction to Smooth Manifolds