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. 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 ( X , T ) {displaystyle (X,{mathcal {T}})} be a topological space and ( Y , d Y ) {displaystyle (Y,d_{Y})} be a metric space. A sequence of functions is said to converge compactly as n → ∞ {displaystyle n o infty } to some function f : X → Y {displaystyle f:X o Y} if, for every compact set K ⊆ X {displaystyle Ksubseteq X} , converges uniformly on K {displaystyle K} as n → ∞ {displaystyle n o infty } . This means that for all compact K ⊆ X {displaystyle Ksubseteq X} ,