Compact-open topology

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