Type of mathematical convergence in topology
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.
Definition
Let
be a topological space and
be a metric space. A sequence of functions
, ![{\displaystyle n\in \mathbb {N} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b98e213fe7ef48da0be47453bc1bb66f37f4eec)
is said to converge compactly as
to some function
if, for every compact set
,
![{\displaystyle f_{n}|_{K}\to f|_{K))](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d95514f8abecc7cfa157c989e934b8acd34900)
uniformly on
as
. This means that for all compact
,
![{\displaystyle \lim _{n\to \infty }\sup _{x\in K}d_{Y}\left(f_{n}(x),f(x)\right)=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d22ddce7e0e5d0c63d73c057252de1e87ff25da)