Editing Proper map (section)

==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]].
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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.
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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>