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Proper map
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In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact.[1] In algebraic geometry, the analogous concept is called a proper morphism.

Definition

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There are several competing definitions of a "proper function". Some authors call a function between two topological spaces proper if the preimage of every compact set in is compact in Other authors call a map 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 is compact. The two definitions are equivalent if is locally compact and Hausdorff.

If is Hausdorff and is locally compact Hausdorff then proper is equivalent to universally closed. A map is universally closed if for any topological space the map is closed. In the case that is Hausdorff, this is equivalent to requiring that for any map the pullback be closed, as follows from the fact that is a closed subspace of

An equivalent, possibly more intuitive definition when and are metric spaces is as follows: we say an infinite sequence of points in a topological space escapes to infinity if, for every compact set only finitely many points are in Then a continuous map is proper if and only if for every sequence of points that escapes to infinity in the sequence escapes to infinity in

Properties

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  • 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 is called a compact covering if for every compact subset there exists some compact subset such that
  • A topological space is compact if and only if the map from that space to a single point is proper.
  • If is a proper continuous map and is a compactly generated Hausdorff space (this includes Hausdorff spaces that are either first-countable or locally compact), then is closed.[2]

Generalization

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

See also

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Citations

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  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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Proper map

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In mathematics, a proper map is a continuous function f:X→Yf: X \to Y between topological spaces XX and YY such that the preimage f−1(K)f^{-1}(K) of every compact subset K⊆YK \subseteq Y is compact in XX. This notion generalizes the idea of compactness from spaces to maps, capturing "compactness at infinity" in a way that ensures well-behaved behavior under limits and preimages. Several equivalent characterizations of proper maps exist, particularly under mild assumptions like Hausdorff or locally compact spaces. For instance, in the context of locally compact Hausdorff spaces, a continuous map is proper if and only if it is closed and has compact fibers. More generally, a map is proper if it is universally closed—meaning that for any space ZZ and map Z→YZ \to Y, the base-changed map Z×YX→ZZ \times_Y X \to Z is closed—and separated (i.e., the induced map on the diagonal is closed). These equivalences were systematically explored by Nicolas Bourbaki in their foundational work on general topology. Key examples of proper maps include closed embeddings (which are proper as they are closed with finite fibers) and projections π:X×K→X\pi: X \times K \to X where KK is compact. Proper maps preserve certain topological properties, such as turning sequences with convergent images into sequences with compact preimages, and they are stable under composition and base change. In non-Hausdorff settings, additional care is needed, as the preimage condition may differ from universal closedness. Proper maps play a crucial role in and its applications, enabling theorems on surjectivity (e.g., proper maps from compact spaces are closed and thus surjective onto their images) and facilitating proofs in , such as the via degree arguments on proper maps. In algebraic geometry, the concept extends to proper morphisms of schemes, which ensure finite and proper base change theorems essential for and .

Definition and Characterizations

Formal Definition

In , a KK of a is called compact if every open cover of KK has a finite subcover. This property captures a form of "boundedness" in abstract topological settings, assuming familiarity with the basic axioms of topological spaces. A f:X→Yf: X \to Y between topological spaces XX and YY is proper if for every K⊆YK \subseteq Y, the preimage f−1(K)f^{-1}(K) is compact in XX. This condition ensures that the map behaves well with respect to in the codomain by pulling it back to the domain. Properness generalizes the notion of compactness to maps, in the sense that it requires the map to preserve compactness "backwards" through preimages. For instance, the unique continuous map from a space XX to a singleton (point) space is proper if and only if XX itself is compact.

Equivalent Conditions

In metric spaces, a continuous map f:X→Yf: X \to Y between separable metric spaces is proper if and only if for every sequence {xn}\{x_n\} in XX that has no convergent subsequence (i.e., diverges to infinity), the image sequence f(xn)f(x_n) has no convergent subsequence in YY. Equivalently, for every sequence {yn}\{y_n\} in YY diverging to infinity, any choice of preimages {xn}⊂f−1(yn)\{x_n\} \subset f^{-1}(y_n) must also diverge to infinity in XX. When XX and YY are locally compact Hausdorff spaces, a continuous f:X→Yf: X \to Y is proper it is a closed map with compact fibers, meaning f−1(y)f^{-1}(y) is compact for every y∈Yy \in Y. A continuous f:X→Yf: X \to Y between s is proper it is universally closed and separated, where universally closed means that for any continuous g:Z→Yg: Z \to Y from another ZZ, the base-changed Z×YX→ZZ \times_Y X \to Z is closed, and separated means the diagonal Δf:X→X×YX\Delta_f: X \to X \times_Y X is a closed immersion. This characterization aligns with the Bourbaki notion of properness for the universally closed part, where the remains closed under arbitrary base changes. If YY is Hausdorff, properness of f:X→Yf: X \to Y implies that fibers f−1(y)f^{-1}(y) are compact for each y∈Yy \in Y, since singletons are compact in YY. Over a compact K⊂YK \subset Y, the preimage f−1(K)f^{-1}(K) is compact, but achieving finite fibers over such KK requires additional assumptions on XX and YY, such as both being locally compact Hausdorff with ff finite-to-one; without these, fibers remain compact but may be infinite.

Properties

Core Properties

Assuming Y is locally compact, a proper map f:X→Yf: X \to Y between topological spaces is closed, meaning that the image of any closed subset of XX is closed in YY. Proper maps also have compact fibers: for every y∈Yy \in Y, the fiber f−1(y)f^{-1}(y) is compact in XX. This property arises directly from the definition, as the singleton {y}\{y\} is compact in YY, so its preimage must be compact under ff. In the context of locally compact Hausdorff spaces, a continuous map is proper if and only if it is closed with compact fibers. Any continuous map from a XX to a YY is proper. Since XX is compact, preimages of compact subsets of YY (which are closed in the Hausdorff space) remain compact, satisfying the properness condition. Conversely, a XX is compact if and only if the unique continuous X→{pt}X \to \{pt\} to a singleton space is proper, as the preimage of the compact point {pt}\{pt\} is all of XX. Assuming Y is locally compact, when f:X→Yf: X \to Y is a surjective proper between Hausdorff spaces, it is a closed map, and YY is the of XX by the whose classes are the compact fibers of ff. This structure implies that ff identifies points within each compact , yielding a compact covering in the sense of a surjection with compact fibers over a Hausdorff base.

Stability Properties

Proper maps exhibit several stability properties with respect to common topological constructions, ensuring that properness is preserved in compositions, pullbacks, and certain products. These properties underscore the robustness of the notion in building more complex spaces while maintaining compactness conditions on preimages. Assuming locally compact Hausdorff spaces where relevant. The class of proper maps is closed under composition. Specifically, if f:X→Yf: X \to Y and g:Y→Zg: Y \to Z are proper maps, then the composition g∘f:X→Zg \circ f: X \to Z is proper. This follows from the stability of universally closed maps under composition and the preservation of separatedness in such settings. Proper s are stable under (or base change). If f:X→Yf: X \to Y is proper and p:Z→Yp: Z \to Y is any continuous , then the pullback f′:X×YZ→Zf': X \times_Y Z \to Z is proper. This stability arises directly from the definition of properness as universally closed and separated, where universal closedness ensures closedness after any base change. Regarding products, the projection πX:X×K→X\pi_X: X \times K \to X is proper whenever KK is a . This holds because compactness of KK implies that the projection is universally closed, and the product inherits separatedness from the spaces involved. For instance, if XX is any and KK is Hausdorff, the fibers over points in XX are homeomorphic to KK, which are compact. In the context of compactly generated Hausdorff spaces, proper maps are necessarily closed maps. That is, if f:X→Yf: X \to Y is proper and YY is compactly generated Hausdorff, then ff maps closed subsets of XX to closed subsets of YY. This extends the core closedness property of proper maps to broader classes of spaces without requiring local compactness of the domain. Finally, if f:X→Yf: X \to Y is a proper onto a YY, then the inverse map f−1:Y→Xf^{-1}: Y \to X is continuous, making ff a . This result leverages the compactness of fibers and the Hausdorff condition to ensure that closed sets in YY map to closed sets in XX under the inverse.

Examples

Canonical Examples

One canonical example of a proper map is the inclusion of a compact subset into a Hausdorff space. Specifically, if KK is a compact subset of a Hausdorff topological space XX, then the i:K→Xi: K \to X is proper because the preimage under ii of any compact subset of XX is either empty or a compact subset of KK, hence compact. Another standard example arises in product spaces: the projection map π:X×K→X\pi: X \times K \to X, where KK is compact and XX is any topological space, is proper. This holds because the preimage of a compact subset C⊆XC \subseteq X is C×KC \times K, and since CC is compact and the product of compact spaces is compact, the preimage is compact. Finite-sheeted covering maps between manifolds provide further illustrations of proper maps. For instance, the double covering map p:S1→S1p: S^1 \to S^1 defined by p(z)=z2p(z) = z^2 for z∈S1⊂Cz \in S^1 \subset \mathbb{C} is proper, as both domain and codomain are compact Hausdorff spaces, making any continuous map between them proper. More generally, any continuous map from a compact space to a Hausdorff space is proper. In metric spaces, proper embeddings of compact sets also exemplify proper maps. The inclusion of a closed ball B‾(0,r)⊂Rn\overline{B}(0, r) \subset \mathbb{R}^n into Rn\mathbb{R}^n is proper, as the closed ball is compact and Rn\mathbb{R}^n is Hausdorff, ensuring preimages of compact sets remain compact. This extends to any compact of a metric space, where the preserves properness via the Hausdorff property.

Counterexamples

A classic counterexample of a continuous that is not proper is the i:(0,1)→Ri: (0,1) \to \mathbb{R}, where (0,1)(0,1) carries the induced from R\mathbb{R}. The preimage under ii of the compact set [0,1]⊂R[0,1] \subset \mathbb{R} is (0,1)(0,1), which is not compact, as it admits the open cover {(1/n,1)∣n∈N}\{(1/n, 1) \mid n \in \mathbb{N}\} with no finite subcover. Another standard example is the projection π:R×R→R\pi: \mathbb{R} \times \mathbb{R} \to \mathbb{R} defined by π(x,y)=x\pi(x,y) = x. This is continuous but not proper, since the preimage π−1([0,1])\pi^{-1}([0,1]) equals [0,1]×R[0,1] \times \mathbb{R}, which is not compact due to unboundedness in the second coordinate. In the context of Euclidean spaces, such projections fail to preserve of preimages even though they are open maps. The universal covering map exp⁡:R→S1\exp: \mathbb{R} \to S^1, given by t↦e2πitt \mapsto e^{2\pi i t}, provides a that is not proper. Here, S1S^1 is compact, but its preimage under exp⁡\exp is all of R\mathbb{R}, which is not compact. This illustrates how infinite-sheeted covering maps generally fail to be proper, as the fibers over points are infinite discrete sets, leading to non-compact total preimages for compact subsets of the base. The i:[Q](/page/Q)→Ri: \mathbb{[Q](/page/Q)} \to \mathbb{R}, where [Q](/page/Q)\mathbb{[Q](/page/Q)} has the , is continuous but not proper. The preimage i−1([0,1])=[Q](/page/Q)∩[0,1]i^{-1}([0,1]) = \mathbb{[Q](/page/Q)} \cap [0,1] is not compact in [Q](/page/Q)\mathbb{[Q](/page/Q)}, as it is not sequentially compact: for example, a of in [0,1][0,1] converging in R\mathbb{R} to 2/2\sqrt{2}/2
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