Definition
A function f : X → Y between two topological spaces is proper if and only if the preimage of every compact set in Y is compact in X.
There are several competing descriptions. For instance, a continuous map f is proper if it is a closed map and the pre-image of every point in Y is compact. For a proof of this fact see the end of this section. More abstractly, f is proper if for any space Z the map
- f × idZ: X × Z → Y × Z
is closed. These definitions are equivalent to the previous one if X is Hausdorff and Y is locally compact Hausdorff.
An equivalent, possibly more intuitive definition is as follows: we say an infinite sequence of points {pi} in a topological space X escapes to infinity if, for every compact set S ⊂ X only finitely many points pi are in S. Then a map f : X → Y is proper if and only if for every sequence of points {pi} that escapes to infinity in X, {f(pi)} escapes to infinity in Y.
This last sequential idea looks like being related to the notion of sequentially proper, see a reference below.
Read more about this topic: Proper Map
Famous quotes containing the word definition:
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)