Compact-open topology

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This article defines a function space topology i.e. a topology on the collection of continuous maps between two topological spaces

Definition

Suppose X and Y are topological spaces. The compact-open topology is a topology we can define on the space of continuous functions C(X,Y) from X to Y as follows.

For a compact subset K \subseteq X and an open subset U \subseteq Y, we define W(K,U) as the set of all continuous maps f:X \to Y such that f(K) \subseteq U. The compact-open topology is the topology with subbasis as the set of all W(K,U)s.

Relation with other function space topologies

Topology Meaning Relationship with compact-open topology
topology of pointwise convergence topology chosen such that a sequence of functions converges iff it converges pointwise; equivalently, the subspace topology inherited from the product topology on the space Y^X of all functions.  ?
topology of uniform convergence
topology of compact convergence

Particular cases

Case for X Case for Y Overall conclusion Necessarily equals topology of pointwise convergence? Necessarily equals topology of uniform convergence? (if that makes sense)
one-point space anything canonically identified with Y, identification is a homeomorphism Yes Yes
discrete space anything canonically identified with Y^X, space of all functions from X to Y, with the product topology Yes Yes (if that makes sense)
anything T0 space also a T0 space: see function space to T0 space is T0 under compact-open topology No No
anything T1 space also a T1 space: see function space to T1 space is T1 under compact-open topology No No
anything Hausdorff space also a Hausdorff space: see function space to Hausdorff space is Hausdorff under compact-open topology
anything regular space also a regular space: see function space to regular space is regular under compact-open topology
anything completely regular space also a completely regular space: see function space to completely regular space is completely regular under compact-open topology