Compact Convergence
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Compact Convergence

In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology.


Let be a topological space and be a metric space. A sequence of functions


is said to converge compactly as to some function if, for every compact set ,

uniformly on as . This means that for all compact ,


  • If and with their usual topologies, with , then converges compactly to the constant function with value 0, but not uniformly.
  • If , and , then converges pointwise to the function that is zero on and one at , but the sequence does not converge compactly.
  • A very powerful tool for showing compact convergence is the Arzelà-Ascoli theorem. There are several versions of this theorem, roughly speaking it states that every sequence of equicontinuous and uniformly bounded maps has a subsequence that converges compactly to some continuous map.


  • If uniformly, then compactly.
  • If is a compact space and compactly, then uniformly.
  • If is locally compact, then compactly if and only if locally uniformly.
  • If is a compactly generated space, compactly, and each is continuous, then is continuous.

See also


  • R. Remmert Theory of complex functions (1991 Springer) p. 95

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