nLab proper map

This entry is about the concept in topology. For variants see at proper morphism.

Idea

In topology, the notion of properness is that of compactness, but generalized from topological spaces to continuous functions between them.

A continuous map $X \to Y$ is proper when “$X$ is compact, relative to $Y$”.

Definitions

There are three inequivalent definitions of a proper map in the literature, of strictly increasing strength.

The first one is the oldest definition, still found in some analysis books.

Definition

A continuous map $f\colon X\to Y$ of topological spaces is proper if it is quasi-proper (alias lax proper): for every compact subset $K\subset Y$, the preimage $f^{-1}K$ is a compact subset of $X$.

The second definition was introduced by Bourbaki to make proper maps closed under base changes.

Definition

A continuous map $f\colon X\to Y$ of topological spaces is proper if it is universally closed: every base change of $f$ is a closed map of topological spaces.

Other equivalent characterizations include the following:

• For every topological space $Z$ the map $f\times id_Z\colon X\times Z\to Y\times Z$ is a closed map. (The original formulation used by Bourbaki.)

• The map $f$ is quasi-proper and closed.

• The map $f$ is closed and every fiber of $f$ is compact.

The third definition was introduced by Grothendieck. It is in the same relation to the previous definition as compact Hausdorff spaces are to compact spaces.

Definition

A continuous map $f\colon X\to Y$ of topological spaces is proper if it is universally closed and separated.

Recall that $f$ is separated if its relative diagonal $X\to X\times_Y X$ is a closed map.

If $Y=\{*\}$ is the terminal topological space, then the first two definitions amount to saying $X$ is compact, whereas the last definition makes $X$ compact and Hausdorff.

If $X$ is a Hausdorff space, then the map $f\colon X\to Y$ is automatically separated, which makes the last two definitions equivalent.

The following criterion explains the necessary and sufficient criterion on a topological space $Y$ that ensures that quasi-proper maps to $Y$ are universally closed.

Proposition

(Whyburn 1965, Palais 1969.) The following properties of a topological space $Y$ are equivalent:

• Every quasi-proper map $X\to Y$ is a closed map;
• Every surjective quasi-proper map $X\to Y$ is a closed map;
• Every injective quasi-proper map $X\to Y$ is a closed map;
• Every compactly closed subset of $Y$ is closed.

If $Y$ is Hausdorff, the last condition is equivalent to $Y$ being a compactly generated space.

Equivalent definitions of universally closed maps

Just like there are various equivalent ways of characterizing compact topological spaces, there are also various equivalent ways of characterizing universally closed maps.

In the following we discuss:

• characterization via nets;

• characterization via intersection of closed sets;

• characterization via closedness;

• characterization as a continuous family of compact fibres;

• further characterizations.

Via nets

Recall that for a topological space $X$ to be compact means for every net $x_\bullet$ in $X$ to have a cluster point in $X$; hence for every net to admits a convergent subnet.

A generalisation of this characterisation to a continuous map $f \colon X \to Y$ is the following:

Definition

A continuous map $f \,\colon\, X \to Y$ is universally closed if and only if for every net $x_\bullet$ in $X$ and for every cluster point $y$ of $f(x_\bullet)$ in $Y$, the net $x_\bullet$ has a cluster point $x \in X$ with $f(x) = y$.

Equivalently:

Such $f$ is universally closed if and only if for every net $x_\bullet \in X$ and every $y \in Y$, if $f(x_\bullet)$ converges to $y$ then $x_\bullet$ admits a subnet converging to $x \in X$ with $f(x) = y$.

Via intersection of closed sets

Recall that, equivalently, a topological space $X$ is compact if and only if for every cofiltered set of non-empty closed subsets $F_i \subset X$, their intersection $\cap_i F_i$ is also non-empty.

Definition

A continuous map $f \colon X \to Y$ is universally closed if and only if

1. $f$ is closed;

2. for every cofiltered set of closed subsets $\{F_i\}_{i \in I}$ of $X$, one has

   $$f(\cap_{i \in I} F_i) = \cap_{i \in I}f(F_i)$$

Via closedness

Recall that, equivalently, a topological space $X$ is compact if and only if the projection map

$$X \times Z \longrightarrow Z$$

out of the product topological space with some $Z$ is a closed map, for every topological space $Z$.

Definition

A continuous map $f \colon X \to Y$ is universally closed if and only if its image under the Cartesian product-functor

$$f \times \mathrm{Id}_Z \colon X \times Z \longrightarrow Y \times Z$$

is a closed map for every topological space $Z$.

As a continuous family of compact spaces

With every function $f \colon X \to Y$, the space $X$ can be described as the union of the fibres $f^{-1}(y)$ with $y \in Y$. The map $f$ is proper when each fibre is compact and the family is continuous in the sense that: for every net $y_\bullet$ converging to $y \in Y$, we ask that the net of sets $f^{-1}(y_\bullet)$ converges to $f^{-1}(y)$. This is equivalent to asking that $f$ be closed.

Definition

A continuous map $f \colon X \to Y$ is universally closed if:

1. $f^{-1}(\{y\})$ is compact for every $y \in Y$;

2. $f$ is closed.

Further characterizations

Proposition

Given a continuous map $f \colon X \to Y$, the following properties are all equivalent:

1. (Def. )

   For every net $x_\bullet$ in $X$ and every cluster point $y$ of $f(x_\bullet)$, the net $x_\bullet$ admits a cluster point $x$ with $f(x) = y$;

2. If $\mathcal{F}$ is a filter on $X$ and if $y$ is a cluster point of $f(\mathcal{F})$, then $\mathcal{F}$ has a cluster point $x \in X$ with $f(x) = y$;

3. (Def. )

   The map $f$ is a closed map and for every cofiltered family $\{F_i\}_{i \in I}$ of closed subsets of $X$, one has $f(\cap_{i \in I} F_i) = \cap_{i \in I}f(F_i)$;

4. (Def. )

   The image $f \times \mathrm{Id}_Z \,\colon\, X \times Z \to Y \times Z$ under the Cartesian product-functor is a closed map for every topological space $Z$;

5. (universally closed)

   For every continuous map $g \colon Z \to Y$, the resulting pullback map

   $$g^\ast(f)\colon X \times_Y Z \to Z$$

   is a closed map;

6. (Def. )

   The map $f$ is a closed map and the inverse image $f^{-1}\big(\{y\}\big)$ of every $y \in Y$ is compact;

7. The map $f$ is a closed map and the inverse image $f^{-1}(K)$ of every compact subspace $K \subset Y$ is compact.

(cf. Bourbaki 1971: Thm. 1 on p. 1 with Def. 1 on p. 97)

Definition

(proper maps)
A continuous function $f \colon X \to Y$ is called universally closed if it satisfies one of the equivalent properties listed in Prop. .

Remark

(Ambiguous terminology)
The notion of compact space is subject to naming ambiguity. For the same notion, some authors will use the term quasi-compact, using compact only when the space is also separated.

For properness the situation is worse as there are three competing definitions. We have defined the one similar to quasi-compact spaces.

In addition one could require $f$ to be separated, that is to require that if a net $x_\bullet$ converges to both $x_1$ and $x_2$ with $f(x_1) = f(x_2)$, then $x_1 = x_2$. This definition of properness resembles the one used in algebraic geometry: see proper morphism. It is also the one to be used in the proper base change theorem.

Finally, some authors use a weaker version of properness (defined above as quasi-properness), where $f \colon X \to Y$ is quasi-proper when $f^{-1}(K)$ is compact for every compact $K \subset Y$. But as explained below, this definition is usually used in situations where such maps are closed.

Further criteria

A quasi-proper map, i.e., a continuous map $f \colon X \to Y$ such that $f^{-1}(K)$ is compact for every compact $K \subset Y$ may not be closed; even when both $X$ and $Y$ are very nice spaces like $\mathbf{T}_5$ spaces.

Example

Let $X$ be an uncountable set and let $p \in X$. Let’s still write $X$ for the discrete topological space associated to it. Let $X_p$ denote the topological space whose underlying set is $X$ but whose opens sets are either all the sets not containing $p$ or the sets containing $p$ with countable complement in $X$.

Then $X_p$ is a $\mathbf{T}_5$ topological space and its compact subsets are all finite. But the identity map

$$X \longrightarrow X_p$$

is not closed.

However this is very often the case in practice: for example when $Y$ is a metric space or a locally compact separated space.

Proposition

Let $f \colon X \to Y$ be a continuous map such that

1. $f^{-1}(K)$ is compact for every compact $K \subset Y$;

2. $Y$ is a $k$-space

then $f$ is closed and thus universally closed.

Also,

Proposition

(maps from compact spaces to separated spaces are proper)

Let $f \colon X \longrightarrow Y$ be a continuous function between topological spaces such that

1. $X$ is compact;

2. $Y$ is separated,

then $f$ is universally closed.

Properties

Proper maps enjoy analogous properties as compact topological spaces do, for example the product of proper maps (with or without the separation condition) is again proper:

Theorem

Let $\{f_i \colon X_i \to Y_i\}_{i \in I}$ be a small indexed family of universally closed maps (Def. ), then their functorial product

$$\prod_{i \in I} f_i \colon \prod_{i \in I} X_i \longrightarrow \prod_{i \in I} Y_i$$

is also a universally closed map. If the maps $f_i$ are separated maps, then so is their product.

Further properties:

• proper maps to locally compact spaces are closed

Comparison with proper maps of locales

Definition

A morphism $f\colon X\to Y$ of locales is universally closed if every base change of $f$ is a closed map of locales, separated if the relative diagonal $X\to X\times_Y X$ is a closed map of locales, and proper if it is universally closed and separated.

These three notions can also be defined categorically by considering $f$ as an object in the category $Loc/Y$ and recalling that the latter category is equvialent to the category of locales in the topos of sheaves of sets on $Y$. Then $f$ satisfies these properties if the corresponding internal locale is compact, separated (meaning the diagonal map is a closed map of locales), or compact and separated, respectively.

Proposition

Every proper map $f\colon X \to Y$ between two topological spaces induces a proper map $\mathrm{Loc}(f)\colon \mathrm{Loc}(X) \to \mathrm{Loc}(Y)$ between the associated locales.

Conversely, if $\mathrm{Loc}(f)\colon \mathrm{Loc}(X) \to \mathrm{Loc}(Y)$ is proper and $Y$ is a $\mathrm{T}_\mathrm{D}$-space, then $f\,:\, X \to Y$ is proper.

Related concepts

• locally proper map

• proper morphism

• open map, closed map

• proper group action

• proper homotopy

• proper equivariant homotopy theory

References

All three definitions are presented in

• Stacks Project, Tag 005M, https://stacks.math.columbia.edu/tag/005M.

Equivalence of the first and second definition is explored in

• Richard S. Palais, When proper maps are closed, Proceedings of the American Mathematical Society 24:4 (1970), 835-836. doi.

One of the early reference on proper maps is

• Nicolas Bourbaki, Topologie Générale (Éléments de Mathématique, Livre III), Chapitres 1 à 2. Second Edition, 1951. Actualites Sci. Ind. 1142, Hermann, Paris.

An expanded treatment is given in the third edition, see Chapter I, Section 10:

• Nicolas Bourbaki, Topologie Générale (Éléments de Mathématique, Livre III), Chapitres 1 à 2. Third Edition, 1961. Actualites Sci. Ind. 1142, Hermann, Paris.

English translation of the 1971 edition:

• Nicolas Bourbaki, General topology, Elements of Mathematics III, Springer (1971, 1990, 1995) [doi:10.1007/978-3-642-61701-0]

The localic version of proper maps was introduced in:

• Peter Johnstone, §1 of: The Gleason cover of a topos, II, Journal of Pure and Applied Algebra 22 3 (1981) 229–247 [doi:10.1016/0022-4049(81)90100-6]

where it is attributed to this “preliminary” (and apparently unpublished) account:

• Peter Johnstone, Factorization and pullback theorems for localic geometric morphisms, Univ. Cath. de Louvain, Sem. de math, pure, Rapport no. 79 (1979)

Additional sources:

• J. J. C. Vermeulen, Proper maps of locales, Journal of Pure and Applied Algebra 92:1 (1994), 79-107. doi.

• M. Korostenski, C. C. A. Labuschagne, Lax proper maps of locales, Journal of Pure and Applied Algebra 208:2 (2007), 655-664. doi.