nLab proper map

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Contents

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

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

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

A continuous map XYX \to Y is proper when “XX is compact, relative to YY”.

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 XX to be compact means for every net x x_\bullet in XX to have a cluster point in XX; hence for every net to admits a convergent subnet.

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

Definition

A continuous map f:XYf \,\colon\, X \to Y is proper iff for every net x x_\bullet in XX and for every cluster point yy of f(x )f(x_\bullet) in YY, the net x x_\bullet has a cluster point xXx \in X with f(x)=yf(x) = y.

Equivalently:

Such ff is proper iff for every net x Xx_\bullet \in X and every yYy \in Y, if f(x )f(x_\bullet) converges to yy then x x_\bullet admits a subnet converging to xXx \in X with f(x)=yf(x) = y.

Via intersection of closed sets

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

Definition

A continuous map f:XYf \,\colon\, X \to Y is proper iff

  1. ff is closed;

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

f( iIF i)= iIf(F i) f(\cap_{i \in I} F_i) = \cap_{i \in I}f(F_i)

Via closedness

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

X×ZZ X \times Z \longrightarrow Z

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

Definition

A continuous map f:XYf \,\colon\, X \to Y is proper iff its image under the Cartesian product-functor

f×Id Z:X×ZY×Z f \times \mathrm{Id}_Z \;\colon\; X \times Z \;\longrightarrow\; Y \times Z

is a closed map for every topological space ZZ.

As a continuous family of compact spaces

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

Definition

A continuous map f:XYf \,\colon\, X \to Y is proper if:

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

  2. ff is closed.

Further characterizations

Proposition

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

  1. (Def. )

    For every net x x_\bullet in XX and every cluster point yy of f(x )f(x_\bullet), the net x x_\bullet admits a cluster point xx with f(x)=yf(x) = y;

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

  3. (Def. )

    ff is a closed map and for every cofiltered family {F i} iI\{F_i\}_{i \in I} of closed subsets of XX, one has f( iIF i)= iIf(F i)f(\cap_{i \in I} F_i) = \cap_{i \in I}f(F_i);

  4. (Def. )

    The image f×Id Z:X×ZY×Zf \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 ZZ;

  5. (universally closed)

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

    g *(f):X× YZZ g^\ast(f) \;\colon\; X \times_Y Z \to Z

    is a closed map;

  6. (Def. )

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

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

Definition

(proper maps)
A continuous function f:XYf \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 ff to be separated, that is to require that if a net x x_\bullet converges to both x 1x_1 and x 2x_2 with f(x 1)=f(x 2)f(x_1) = f(x_2), then x 1=x 2x_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:XYf \colon X \to Y is proper when f 1(K)f^{-1}(K) is compact for every compact KYK \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:XYf \colon X \to Y such that f 1(K)f^{-1}(K) is compact for every compact KYK \subset Y may not be closed; even when both XX and YY are very nice spaces like T 5\mathbf{T}_5 spaces.

Example

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

Then X pX_p is a T 5\mathbf{T}_5 topological space and its compact subsets are all finite. But the identity map

XX p X \longrightarrow X_p

is not closed.

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

Proposition

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

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

  2. YY is a k k -space

then ff is closed and thus proper.

Also,

Proposition

(maps from compact spaces to separated spaces are proper)

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

  1. XX is compact;

  2. YY is separated,

then ff 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:X iY i} iI\{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

iIf i: iIX i iIY i \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:XYf\,:\, X \to Y between two topological spaces induces a proper map Loc(f):Loc(X)Loc(Y)\mathrm{Loc}(f)\,:\, \mathrm{Loc}(X) \to \mathrm{Loc}(Y) between the associated locales.

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

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:

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)

Last revised on February 11, 2024 at 12:12:16. See the history of this page for a list of all contributions to it.