- Proper map
In
mathematics , acontinuous function betweentopological space s is called proper ifinverse image s of compact subsets are compact. Inalgebraic geometry , the analogous concept is called aproper morphism .Definition
A function "f" : "X" → "Y" between two
topological space s is proper if and only if thepreimage of every compact set in "Y" is compact in "X".There are several competing descriptions. For instance, a continuous map "f" is proper if it is a
closed map and the pre-image of every point in "Y" is compact. For a proof of this fact see the end of this section. More abstractly, "f" is proper if it is a closed map, and for any space "Z" the"Z"): "X" × "Z" → "Y" × "Z"is closed. These definitions are equivalent to the previous one if the space "X" is
locally compact .An equivalent, possibly more intuitive definition is as follows: we say an infinite sequence of points {"p""i"} in a topological space "X" escapes to infinity if, for every compact set "S" ⊂ "X" only finitely many points "p""i" are in "S". Then a map "f" : "X" → "Y" is proper if and only if for every sequence of points {"p""i"} that escapes to infinity in "X", {"f"("p""i")} escapes to infinity in "Y".
Proof of fact
Let f: X o Y be a continuous closed map, such that f^{-1}(y) is compact (in X) for all y in Y. Let K be a compact subset of Y. We will show that f^{-1}(K) is compact.
Let U_{lambda} vert lambda in Lambda } 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 all k in K there is a finite set gamma_k subset Lambda such that f^{-1}(k) subset cup_{lambda in gamma_k} U_{lambda}.The set X setminus cup_{lambda in gamma_k} U_{lambda} is closed. Its image is closed in Y, because f is a closed map. Hence the set
V_k = Y setminus f(X setminus cup_{lambda in gamma_k} U_{lambda}) is open in Y. It is easy to check that V_k contains the point k.Now K subset 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 subset 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, thus Gamma is finite.
Now it follows that f^{-1}(K) subset f^{-1}(cup_{i=1}^s V_{k_i}) subset cup_{lambda in Gamma} U_{lambda} and we have found a finite subcover of f^{-1}(K), which completes the proof.
Properties
*A topological space is compact if and only if the map from that space to a single point is proper.
*Every continuous map from a compact space to aHausdorff space is both proper and closed.
*If "f" : "X" → "Y" is a proper continuous map and "Y" is acompactly generated Hausdorff space (this includes Hausdorff spaces which are eitherfirst-countable orlocally compact ), then "f" is closed.Generalization
It is possible to generalize the notion of proper maps of topological spaces to locales and topoi, see Harv|Johnston|2002.
See also
*
Perfect map
*Topology glossary References
* | year=1998
*, esp. section C3.2 "Proper maps"
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