This entry is about the concept in topology. For variants see at proper morphism.
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
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 $Y$”.
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:
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:
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 $f(x) = y$.
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.
A continuous map $f \,\colon\, X \to Y$ is proper iff
$f$ is closed;
for every cofiltered set of closed subsets $\{F_i\}_{i \in I}$ of $X$, one has
Recall that, equivalently, a topological space $X$ is compact if and only if the projection map
out of the product topological space with some $Z$ is a closed map, for every topological space $Z$.
A continuous map $f \,\colon\, X \to Y$ is proper iff its image under the Cartesian product-functor
is a closed map for every topological space $Z$.
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.
A continuous map $f \,\colon\, X \to Y$ is proper if:
$f^{-1}(\{y\})$ is compact for every $y \in Y$;
$f$ is closed.
Given a continuous map $f \colon X \to Y$, the following properties are all equivalent:
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 $f(x) = y$;
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 $f(x) = y$;
$f$ is a closed map and for every cofiltered family $\{F_i\}_{i \in I}$ of closed subsets of $X$, one has $f(\cap_{i \in I} F_i) = \cap_{i \in I}f(F_i)$;
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 $Z$;
For every continuous map $g \colon Z \to Y$, the resulting pullback map
is a closed map;
$f$ is a closed map and the inverse image $f^{-1}\big(\{y\}\big)$ of every $y \in Y$ is compact;
$f$ is a closed map and the inverse image $f^{-1}(K)$ of every compact subspace $K \subset Y$ is compact.
(proper maps)
A continuous function $f \colon X \to Y$ is called proper if it satisfies one of the equivalent properties listed in Prop. .
(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.
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.
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
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.
Let $f \colon X \to Y$ be a continuous map such that
$f^{-1}(K)$ is compact for every compact $K \subset Y$;
$Y$ is a $k$-space
then $f$ is closed and thus proper.
Also,
(maps from compact spaces to separated spaces are proper)
Let $f \colon X \longrightarrow Y$ be a continuous function between topological spaces such that
then $f$ is proper.
Proper maps enjoy analogous properties as compact topological spaces do, for example the product of proper maps is again proper:
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
is also a proper map.
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.
One of the early reference on proper maps is
A considerable expanded treatment is given in the third edition, see Chapter I, Section 10:
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:
Last revised on February 11, 2024 at 12:12:16. See the history of this page for a list of all contributions to it.