Compact-Open Topology

The compact-open topology is a common topology used on function spaces. Suppose and are topological spaces and is the set of continuous maps . The compact-open topology on is generated by subsets of the following form,

(1)

where is compact in and is open in . (Hence the terminology "compact-open.") It is important to note that these sets are not closed under intersection, and do not form a topological basis. Instead, the sets form a subbasis for the compact-open topology. That is, the open sets in the compact-open topology are the arbitrary unions of finite intersections of .

The simplest function space to compare topologies is the function space of continuous real-valued functions . A sequence of functions converges to iff every containing contains all but a finite number of the . Hence, for all and all , there exists an such that for all ,

(2)

For example, the sequence of functions converges to the zero function, although each function is unbounded.

When is a metric space, the compact-open topology is the same as the topology of uniform convergence on compact sets. If is a locally compact T₂-space, a fairly weak condition, then the evaluation map

(3)

defined by is continuous. Similarly, is continuous iff the map , given by , is continuous. Hence, the compact-open topology is the right topology to use in homotopy theory.