Compact convergence

Examples

• If ${\displaystyle X=(0,1)\subseteq \mathbb{R}}$ and ${\displaystyle Y=\mathbb{R}}$ with their usual topologies, with ${\displaystyle f_n(x):=x^n}$, then ${\displaystyle f_n}$ converges compactly to the constant function with value 0, but not uniformly.
• If ${\displaystyle X=(0,1]}$, ${\displaystyle Y=\mathbb{R}}$ and ${\displaystyle f_n(x)=x^n}$, then ${\displaystyle f_n}$ converges pointwise to the function that is zero on ${\displaystyle (0,1)}$ and one at ${\displaystyle 1}$, 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.

Properties

• If ${\displaystyle f_n\to f}$ uniformly, then ${\displaystyle f_n\to f}$ compactly.
• If ${\displaystyle (X,\mathcal{T})}$ is a compact space and ${\displaystyle f_n\to f}$ compactly, then ${\displaystyle f_n\to f}$ uniformly.
• If ${\displaystyle (X,\mathcal{T})}$ is a locally compact space, then ${\displaystyle f_n\to f}$ compactly if and only if ${\displaystyle f_n\to f}$ locally uniformly.
• If ${\displaystyle (X,\mathcal{T})}$ is a compactly generated space, ${\displaystyle f_n\to f}$ compactly, and each ${\displaystyle f_n}$ is continuous, then ${\displaystyle f}$ is continuous.