• Source: Proper map

In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.




Definition


There are several competing definitions of a "proper function".
Some authors call a function



f
:
X
→
Y


{\displaystyle f:X\to Y}

between two topological spaces proper if the preimage of every compact set in



Y


{\displaystyle Y}

is compact in



X
.


{\displaystyle X.}


Other authors call a map



f


{\displaystyle f}

proper if it is continuous and closed with compact fibers; that is if it is a continuous closed map and the preimage of every point in



Y


{\displaystyle Y}

is compact. The two definitions are equivalent if



Y


{\displaystyle Y}

is locally compact and Hausdorff.

If



X


{\displaystyle X}

is Hausdorff and



Y


{\displaystyle Y}

is locally compact Hausdorff then proper is equivalent to universally closed. A map is universally closed if for any topological space



Z


{\displaystyle Z}

the map



f
×

id

Z


:
X
×
Z
→
Y
×
Z


{\displaystyle f\times \operatorname {id} _{Z}:X\times Z\to Y\times Z}

is closed. In the case that



Y


{\displaystyle Y}

is Hausdorff, this is equivalent to requiring that for any map



Z
→
Y


{\displaystyle Z\to Y}

the pullback



X

×

Y


Z
→
Z


{\displaystyle X\times _{Y}Z\to Z}

be closed, as follows from the fact that



X

×

Y


Z


{\displaystyle X\times _{Y}Z}

is a closed subspace of



X
×
Z
.


{\displaystyle X\times Z.}


An equivalent, possibly more intuitive definition when



X


{\displaystyle X}

and



Y


{\displaystyle Y}

are metric spaces is as follows: we say an infinite sequence of points



{

p

i


}


{\displaystyle \{p_{i}\}}

in a topological space



X


{\displaystyle X}

escapes to infinity if, for every compact set



S
⊆
X


{\displaystyle S\subseteq X}

only finitely many points




p

i




{\displaystyle p_{i}}

are in



S
.


{\displaystyle S.}

Then a continuous map



f
:
X
→
Y


{\displaystyle f:X\to Y}

is proper if and only if for every sequence of points




{

p

i


}



{\displaystyle \left\{p_{i}\right\}}

that escapes to infinity in



X
,


{\displaystyle X,}

the sequence




{

f

(

p

i


)


}



{\displaystyle \left\{f\left(p_{i}\right)\right\}}

escapes to infinity in



Y
.


{\displaystyle Y.}





Properties


Every continuous map from a compact space to a Hausdorff space is both proper and closed.
Every surjective proper map is a compact covering map.
A map



f
:
X
→
Y


{\displaystyle f:X\to Y}

is called a compact covering if for every compact subset



K
⊆
Y


{\displaystyle K\subseteq Y}

there exists some compact subset



C
⊆
X


{\displaystyle C\subseteq X}

such that



f
(
C
)
=
K
.


{\displaystyle f(C)=K.}


A topological space is compact if and only if the map from that space to a single point is proper.
If



f
:
X
→
Y


{\displaystyle f:X\to Y}

is a proper continuous map and



Y


{\displaystyle Y}

is a compactly generated Hausdorff space (this includes Hausdorff spaces that are either first-countable or locally compact), then



f


{\displaystyle f}

is closed.




Generalization


It is possible to generalize
the notion of proper maps of topological spaces to locales and topoi, see (Johnstone 2002).




See also


Almost open map – Map that satisfies a condition similar to that of being an open map.
Open and closed maps – A function that sends open (resp. closed) subsets to open (resp. closed) subsets
Perfect map – Continuous closed surjective map, each of whose fibers are also compact sets
Topology glossary




Citations






References

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