# nLab proper map

Contents

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

# Contents

## Idea

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

## 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:

### 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 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. )

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 $Z$;

4. (Def. )

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

5. 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. $f$ 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.

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:

The localic version of proper maps was introduced in:

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)