Exhaustion by compact sets

In mathematics, especially general topology and analysis, an exhaustion by compact sets[1] 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. ${\displaystyle K_{i}\subseteq {\text{int}}(K_{i+1})}$ for each ${\displaystyle i}$ and ${\displaystyle X=\bigcup _{i=1}^{\infty }K_{i}}$. A space admitting an exhaustion by compact sets is called exhaustible by compact sets.

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

Occasionally some authors drop the requirement that ${\displaystyle K_{i}}$ is in the interior of ${\displaystyle K_{i+1}}$, but then the property becomes the same as the space being σ-compact, namely a countable union of compact subsets.