Exhaustion by compact sets

In mathematics, especially analysis, exhaustion by compact sets of a topological space ${\displaystyle X}$ is a nested sequence of compact subsets ${\displaystyle K_{i}}$ of ${\displaystyle X}$ (i.e. ${\displaystyle K_{1}\subseteq K_{2}\subseteq K_{3}\subseteq \cdots }$), such that ${\displaystyle K_{i}}$ is contained in the interior of ${\displaystyle K_{i+1}}$ , i.e. ${\displaystyle K_{i}\subset {\text{int}}(K_{i+1})}$ for each ${\displaystyle i}$ and ${\displaystyle X=\bigcup _{i=1}^{\infty }K_{i}}$ .

For example, consider ${\displaystyle X={\mathbb {R} }^{n}}$ and the sequence of closed balls ${\displaystyle K_{i}=\{x:|x|\leq i\}}$.