Definition
Let X and Y be two topological spaces, and let C(X,Y) denote the set of all continuous maps between X and Y. Given a compact subset K of X and an open subset U of Y, let V(K,U) denote the set of all functions ƒ ∈ C(X,Y) such that ƒ(K) ⊂ U. Then the collection of all such V(K,U) is a subbase for the compact-open topology on C(X,Y). (This collection does not always form a base for a topology on C(X,Y).)
When working in the category of compactly-generated spaces, it is common to modify this definition by restricting to the subbase formed from those K which are the image of a compact Hausdorff space. Of course, if X is compactly generated and Hausdorff, this definition coincides with the previous one. However, the modified definition is crucial if one wants the convenient category of compactly-generated weak Hausdorff spaces to be Cartesian closed, among other useful properties. The confusion between this definition and the one above is caused by differing usage of the word compact.
Read more about this topic: Compact-open Topology
Famous quotes containing the word definition:
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (17721834)
“Im beginning to think that the proper definition of Man is an animal that writes letters.”
—Lewis Carroll [Charles Lutwidge Dodgson] (18321898)