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)
|
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)
|
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)
|
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.
