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 $Y$”.

Definition

Just like there are various equivalent ways of characterizing compact topological spaces, there are also various equivalent ways of characterizing proper functions.

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 proper iff 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 proper iff 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 proper iff

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 proper iff 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 ask that $f$ be closed.

Definition

A continuous map $f \,\colon\, X \to Y$ is proper 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. )

   $f$ 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. )

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

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

Definition

(proper maps)
A continuous function $f \colon X \to Y$ is called proper 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, where $f \colon X \to Y$ is 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 these maps are always closed.

Further criteria

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 proper.

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 proper.

Properties

Proper maps enjoy analogous properties as compact topological spaces do, for example the product of proper maps is again proper:

Theorem

Let $\{f_i \colon X_i \to Y_i\}_{i \in I}$ be a small indexed family of continuous proper 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 proper map.

Comparison with proper maps of locales

Proposition

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

Conversely, if $\mathrm{Loc}(f)\,:\, \mathrm{Loc}(X) \to \mathrm{Loc}(Y)$ 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

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.

A considerable 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)