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Proper map

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(Redirected from Proper function) Map between topological spaces with the property that the preimage of every compact is compact This article is about the concept in topology. For the concept in convex analysis, see proper convex function.

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 → Y {\displaystyle f:X\to Y} between two topological spaces proper if the preimage of every compact 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 continuous closed map and the preimage of every point in Y {\displaystyle Y} is compact. The two definitions are equivalent if Y {\displaystyle Y} is locally compact and Hausdorff.

Partial proof of equivalence

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.

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.

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.

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.}

An equivalent, possibly more intuitive definition when X {\displaystyle X} and Y {\displaystyle Y} are metric spaces 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.}

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 → 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.}
  • A topological space is compact if and only if the map from that space to a single point is proper.
  • If f : X → Y {\displaystyle f:X\to Y} is a proper continuous map and Y {\displaystyle Y} is a compactly generated Hausdorff space (this includes Hausdorff spaces that are either first-countable or locally compact), then f {\displaystyle f} is closed.

Generalization

It is possible to generalize the notion of proper maps of topological spaces to locales and topoi, see (Johnstone 2002).

See also

Citations

  1. Lee 2012, p. 610, above Prop. A.53.
  2. 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.

References

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