Compact convergence

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

Let ${\displaystyle (X,{\mathcal {T}})}$ be a topological space and ${\displaystyle (Y,d_{Y})}$ be a metric space. A sequence of functions

is said to converge compactly as ${\displaystyle n\to \infty }$ to some function ${\displaystyle f:X\to Y}$ if, for every compact set ${\displaystyle K\subseteq X}$,

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