The compact-open topology is a common topology used on function spaces. Suppose and are topological spaces and is the set of continuous maps from . The compact-open topology on is generated by subsets of the following form,
(1)
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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 space of real-valued continuous functions . A sequence of functions converges to iff for every containing contains all but a finite number of the . Hence, for all and all , there exists an such that for all ,
(2)
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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 compact convergence. If is a locally compact T2-space, a fairly weak condition, then the evaluation map
(3)
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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.