Proper map

<h2 id="definition">Definition</h2>
<p>There are several competing definitions of a "proper <a href="/facts/Function_(mathematics)/CdReEKCh">function</a>". 
Some authors call a function 
  
    
      
        f
        :
        X
        →
        Y
      
    
    {\displaystyle f:X\to Y}
  
 between two <a href="/facts/Topological_space/v2ut7CbK">topological spaces</a> proper if the <a href="/facts/Preimage/KANefMAl">preimage</a> of every <a href="/facts/Compact_space/d0cgXJH7">compact</a> set in 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 is compact in 
  
    
      
        X
        .
      
    
    {\displaystyle X.}

Other authors call a map 
  
    
      
        f
      
    
    {\displaystyle f}
  
 proper if it is continuous and closed with compact fibers; that is if it is a <a href="/facts/Continuous_map/sKbl02pB">continuous</a> <a href="/facts/Closed_map/abHUKX7l">closed map</a> and the preimage of every point in 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 is <a href="/facts/Compact_set/d0cgXJH7">compact</a>. The two definitions are equivalent if 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 is <a href="/facts/Locally_compact_space/hesalT7q">locally compact</a> and <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a>.

</p>

<table><tbody><tr><th>Partial proof of equivalence</th></tr><tr><td><p>Let                     f        :        X        →        Y              {\displaystyle f:X\to Y}   be a closed map, such that                               f                      −            1                          (        y        )              {\displaystyle f^{-1}(y)}   is compact (in                     X              {\displaystyle X}  ) for all                     y        ∈        Y        .              {\displaystyle y\in Y.}   Let                     K              {\displaystyle K}   be a compact subset of                     Y        .              {\displaystyle Y.}   It remains to show that                               f                      −            1                          (        K        )              {\displaystyle f^{-1}(K)}   is compact.</p><p>Let                               {                                    U                              a                                      :            a            ∈            A                    }                      {\displaystyle \left\{U_{a}:a\in A\right\}}   be an open cover of                               f                      −            1                          (        K        )        .              {\displaystyle f^{-1}(K).}   Then for all                     k        ∈        K              {\displaystyle k\in K}   this is also an open cover of                               f                      −            1                          (        k        )        .              {\displaystyle f^{-1}(k).}   Since the latter is assumed to be compact, it has a finite subcover. In other words, for every                     k        ∈        K        ,              {\displaystyle k\in K,}   there exists a finite subset                               γ                      k                          ⊆        A              {\displaystyle \gamma _{k}\subseteq A}   such that                               f                      −            1                          (        k        )        ⊆                  ∪                      a            ∈                          γ                              k                                                              U                      a                          .              {\displaystyle f^{-1}(k)\subseteq \cup _{a\in \gamma _{k}}U_{a}.}  The set                     X        ∖                  ∪                      a            ∈                          γ                              k                                                              U                      a                                {\displaystyle X\setminus \cup _{a\in \gamma _{k}}U_{a}}   is closed in                     X              {\displaystyle X}   and its image under                     f              {\displaystyle f}   is closed in                     Y              {\displaystyle Y}   because                     f              {\displaystyle f}   is a closed map. Hence the set                              V                      k                          =        Y        ∖        f                  (                      X            ∖                          ∪                              a                ∈                                  γ                                      k                                                                                      U                              a                                              )                      {\displaystyle V_{k}=Y\setminus f\left(X\setminus \cup _{a\in \gamma _{k}}U_{a}\right)}   is open in                     Y        .              {\displaystyle Y.}   It follows that                               V                      k                                {\displaystyle V_{k}}   contains the point                     k        .              {\displaystyle k.}  Now                     K        ⊆                  ∪                      k            ∈            K                                    V                      k                                {\displaystyle K\subseteq \cup _{k\in K}V_{k}}   and because                     K              {\displaystyle K}   is assumed to be compact, there are finitely many points                               k                      1                          ,        …        ,                  k                      s                                {\displaystyle k_{1},\dots ,k_{s}}   such that                     K        ⊆                  ∪                      i            =            1                                s                                    V                                    k                              i                                                    .              {\displaystyle K\subseteq \cup _{i=1}^{s}V_{k_{i}}.}   Furthermore, the set                     Γ        =                  ∪                      i            =            1                                s                                    γ                                    k                              i                                                          {\displaystyle \Gamma =\cup _{i=1}^{s}\gamma _{k_{i}}}   is a finite union of finite sets, which makes                     Γ              {\displaystyle \Gamma }   a finite set.</p><p>Now it follows that                               f                      −            1                          (        K        )        ⊆                  f                      −            1                                    (                                    ∪                              i                =                1                                            s                                                    V                                                k                                      i                                                                                )                ⊆                  ∪                      a            ∈            Γ                                    U                      a                                {\displaystyle 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        )        ,              {\displaystyle f^{-1}(K),}   which completes the proof.</p></td></tr></tbody></table>
<p>If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is Hausdorff and 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 is locally compact Hausdorff then proper is equivalent to universally closed. A map is universally closed if for any topological space 
  
    
      
        Z
      
    
    {\displaystyle Z}
  
 the map 
  
    
      
        f
        ×
        
          id
          
            Z
          
        
        :
        X
        ×
        Z
        →
        Y
        ×
        Z
      
    
    {\displaystyle f\times \operatorname {id} _{Z}:X\times Z\to Y\times Z}
  
 is closed. In the case that 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 is Hausdorff, this is equivalent to requiring that for any map 
  
    
      
        Z
        →
        Y
      
    
    {\displaystyle Z\to Y}
  
 the pullback 
  
    
      
        X
        
          ×
          
            Y
          
        
        Z
        →
        Z
      
    
    {\displaystyle X\times _{Y}Z\to Z}
  
 be closed, as follows from the fact that 
  
    
      
        X
        
          ×
          
            Y
          
        
        Z
      
    
    {\displaystyle X\times _{Y}Z}
  
 is a closed subspace of 
  
    
      
        X
        ×
        Z
        .
      
    
    {\displaystyle X\times Z.}

</p><p>An equivalent, possibly more intuitive definition when 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 are <a href="/facts/Metric_space/gnBBRXtc">metric spaces</a> is as follows: we say an infinite sequence of points 
  
    
      
        {
        
          p
          
            i
          
        
        }
      
    
    {\displaystyle \{p_{i}\}}
  
 in a topological space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 escapes to infinity if, for every compact set 
  
    
      
        S
        ⊆
        X
      
    
    {\displaystyle S\subseteq X}
  
 only finitely many points 
  
    
      
        
          p
          
            i
          
        
      
    
    {\displaystyle p_{i}}
  
 are in 
  
    
      
        S
        .
      
    
    {\displaystyle S.}
  
 Then a continuous map 
  
    
      
        f
        :
        X
        →
        Y
      
    
    {\displaystyle f:X\to Y}
  
 is proper if and only if for every sequence of points 
  
    
      
        
          {
          
            p
            
              i
            
          
          }
        
      
    
    {\displaystyle \left\{p_{i}\right\}}
  
 that escapes to infinity in 
  
    
      
        X
        ,
      
    
    {\displaystyle X,}
  
 the sequence 
  
    
      
        
          {
          
            f
            
              (
              
                p
                
                  i
                
              
              )
            
          
          }
        
      
    
    {\displaystyle \left\{f\left(p_{i}\right)\right\}}
  
 escapes to infinity in 
  
    
      
        Y
        .
      
    
    {\displaystyle Y.}

</p>
<h2 id="properties">Properties</h2>
<ul><li>Every continuous map from a compact space to a <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff space</a> is both proper and <a href="/facts/Closed_map/abHUKX7l">closed</a>.</li>
<li>Every <a href="/facts/Surjective_map/KezhLpCb">surjective</a> proper map is a compact covering map.
<ul><li>A map 
  
    
      
        f
        :
        X
        →
        Y
      
    
    {\displaystyle f:X\to Y}
  
 is called a compact covering if for every compact subset 
  
    
      
        K
        ⊆
        Y
      
    
    {\displaystyle K\subseteq Y}
  
 there exists some compact subset 
  
    
      
        C
        ⊆
        X
      
    
    {\displaystyle C\subseteq X}
  
 such that 
  
    
      
        f
        (
        C
        )
        =
        K
        .
      
    
    {\displaystyle f(C)=K.}
  
</li></ul></li>
<li>A topological space is compact if and only if the map from that space to a single point is proper.</li>
<li>If 
  
    
      
        f
        :
        X
        →
        Y
      
    
    {\displaystyle f:X\to Y}
  
 is a proper continuous map and 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 is a <a href="/facts/Compactly_generated_Hausdorff_space/2A5umYGa">compactly generated Hausdorff space</a> (this includes Hausdorff spaces that are either <a href="/facts/First-countable_space/HDuTYa6A">first-countable</a> or <a href="/facts/Locally_compact_space/hesalT7q">locally compact</a>), then 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is closed.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></li></ul>
<h2 id="generalization">Generalization</h2>
<p>It is possible to generalize 
the notion of proper maps of topological spaces to <a href="/facts/Pointless_topology/AASaqh3w">locales</a> and <a href="/facts/Topos/zzttCIwC">topoi</a>, see (Johnstone 2002).
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Almost_open_map/ta3sk8AY">Almost open map</a> – Map that satisfies a condition similar to that of being an open map.</li>
<li><a href="/facts/Open_and_closed_maps/abHUKX7l">Open and closed maps</a> – A function that sends open (resp. closed) subsets to open (resp. closed) subsets</li>
<li><a href="/facts/Perfect_map/BSwIMkwK">Perfect map</a> – Continuous closed surjective map, each of whose fibers are also compact sets</li>
<li><a href="/facts/Topology_glossary/DVkS2W7J">Topology glossary</a></li></ul>
<h2 id="citations">Citations</h2>

<ul><li><a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Bourbaki, Nicolas</a> (1998). <i>General topology. Chapters 5–10</i>. Elements of Mathematics. Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-64563-4. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1726872">1726872</a>.</li>
<li><a href="/facts/Peter_Johnstone_(mathematician)/j3k2A3Xl">Johnstone, Peter</a> (2002). <i>Sketches of an elephant: a topos theory compendium</i>. Oxford: <a href="/facts/Oxford_University_Press/CYyTzzWG">Oxford University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-851598-7., esp. section C3.2 "Proper maps"</li>
<li><a href="/facts/Ronald_Brown_(mathematician)/mT6ArHj0">Brown, Ronald</a> (2006). <i>Topology and groupoids</i>. North Carolina: <a href="/facts/Booksurge/Imzk2wKk">Booksurge</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-4196-2722-8., esp. p. 90 "Proper maps" and the Exercises to Section 3.6.</li>
<li><a href="/facts/Ronald_Brown_(mathematician)/mT6ArHj0">Brown, Ronald</a> (1973). "Sequentially proper maps and a sequential compactification". <i><a href="/facts/Journal_of_the_London_Mathematical_Society/jdcf3vhX">Journal of the London Mathematical Society</a></i>. Second series. 7 (3): 515–522. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2Fjlms%2Fs2-7.3.515">10.1112/jlms/s2-7.3.515</a>.</li>
<li><a href="/facts/John_M._Lee/LBwVxcC2">Lee, John M.</a> (2012). <i>Introduction to Smooth Manifolds</i>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>. Vol. 218 (Second ed.). New York London: <a href="/facts/Springer_Science%2BBusiness_Media/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4419-9981-8. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/808682771">808682771</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Lee 2012, p. 610, above Prop. A.53. - Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol. 218 (Second ed.). New York London: Springer-Verlag. ISBN 978-1-4419-9981-8. OCLC 808682771. <a href="https://search.worldcat.org/oclc/808682771" target="_blank">https://search.worldcat.org/oclc/808682771</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Palais, Richard S. (1970). "When proper maps are closed". Proceedings of the American Mathematical Society. 24 (4): 835–836. doi:10.1090/s0002-9939-1970-0254818-x. MR 0254818. <a href="/wiki/Richard_Palais" target="_blank">/wiki/Richard_Palais</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Proper map open-in-new

Proper map