# Compact convergence: Wikis

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# Encyclopedia

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.

## Definition

Let $(X_{1}, \mathcal{T}_{1})$ be a topological space and (X2,d2) be a metric space. A sequence of functions

$f_{n} : X_{1} \to X_{2}$, $n \in \mathbb{N},$

is said to converge compactly as $n \to \infty$ to some function $f : X_{1} \to X_{2}$ if, for every compact set $K \subseteq X_{1}$,

$(f_{n})|_{K} \to f|_{K}$

converges uniformly on K as $n \to \infty$. This means that for all compact $K \subseteq X_{1}$,

$\lim_{n \to \infty} \sup_{x \in K} d_{2} \left( f_{n} (x), f(x) \right) = 0.$

## Examples

• If $X_{1} = (0, 1) \subset \mathbb{R}$ and $X_{2} = \mathbb{R}$ with their usual topologies, with fn(x): = xn, then fn converges compactly to the constant function with value 0, but not uniformly.
• If X1 = (0,1], $X_2=\R$ and fn(x) = xn, then fn converges pointwise to the function that is zero on (0,1) and one at 1, but the sequence does not converge compactly.

## Properties

• If $f_{n} \to f$ uniformly, then $f_{n} \to f$ compactly.
• If $f_{n} \to f$ compactly and $(X_{1}, \mathcal{T}_{1})$ is itself a compact space, then $f_{n} \to f$ uniformly.
• If $f_n\to f$ compactly and each fn is continuous, then f is continuous.