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compact-open topology (Definition)

Let $ X$ and $ Y$ be topological spaces, and let $ C(X,Y)$ be the set of continuous maps from $ X$ to $ Y.$ Given a compact subspace $ K$ of $ X$ and an open set $ U$ in $ Y,$ let

$\displaystyle {\mathcal U}_{K,U} := \left\{f\in C(X,Y):\: f(x)\in U\, \text{whenever}\, x\in K\right\}. $

Define the compact-open topology on $ C(X,Y)$ to be the topology generated by the subbasis

$\displaystyle \left\{{\mathcal U}_{K,U}:\: K\subset X\,\text{compact,}\quad U\subset Y\, \text{open} \right\}. $

If $ Y$ is a uniform space (for example, if $ Y$ is a metric space), then this is the topology of uniform convergence on compact sets. That is, a sequence $ \left(f_n\right)$ converges to $ f$ in the compact-open topology if and only if for every compact subspace $ K$ of $ X,$ $ \left(f_n\right)$ converges to $ f$ uniformly on $ K$. If in addition $ X$ is a compact space, then this is the topology of uniform convergence.



"compact-open topology" is owned by antonio.
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See Also: uniform convergence

Other names:  topology of compact convergence
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Cross-references: addition, converges, sequence, compact sets, uniform convergence, metric space, uniform space, subbasis, generated by, open set, subspace, compact, continuous maps, topological spaces
There are 10 references to this entry.

This is version 5 of compact-open topology, born on 2003-02-05, modified 2003-02-07.
Object id is 3976, canonical name is CompactOpenTopology.
Accessed 5998 times total.

Classification:
AMS MSC: 54-00 (General topology :: General reference works )
Pending Errata and Addenda
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References by jocaps on 2007-12-08 19:57:36
Hi,

Could you please post reference where it is shown that the topology coincide with the topology of uniform convergence in the case X is compact and Y is a uniform space. Thanks
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