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Compact-Open Topology


The compact-open topology is a common topology used on function spaces. Suppose X and Y are topological spaces and C(X,Y) is the set of continuous maps from f:X->Y. The compact-open topology on C(X,Y) is generated by subsets of the following form,

 B(K,U)={f|f(K) subset U},
(1)

where K is compact in X and U is open in Y. (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 B(K,U) 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 B(K,U).

Compact-open topology

The simplest function space to compare topologies is the space of real-valued continuous functions f:R->R. A sequence of functions f_n converges to f=0 iff for every B(K,U) containing f contains all but a finite number of the f_n. Hence, for all K>0 and all epsilon>0, there exists an N such that for all n>N,

 |f_n(x)|<epsilon for all |x|<=K.
(2)

For example, the sequence of functions f_n=sin(nx/2)/(n+1)+x^(2n)/e^(-n^2/2) converges to the zero function, although each function is unbounded.

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

 e:X×C(X,Y)->Y
(3)

defined by e(x,f)=f(x) is continuous. Similarly, H:X×Z->Y is continuous iff the map H^~:Z->C(X,Y), given by H(x,z)=H^~(z)(x), is continuous. Hence, the compact-open topology is the right topology to use in homotopy theory.


See also

Algebraic Topology, Compact Convergence, Homotopy Theory, Topological Space

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Compact-Open Topology." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Compact-OpenTopology.html

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