Exhaustion by compact sets

In mathematics, especially general topology and analysis, an 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 each ${\displaystyle K_{i}}$ is contained in the interior of ${\displaystyle K_{i+1}}$, i.e. ${\displaystyle K_{i}\subset {\text{int}}(K_{i+1})}$, and ${\displaystyle X=\bigcup _{i=1}^{\infty }K_{i}}$.

A space admitting an exhaustion by compact sets is called exhaustible by compact sets.

As an example, for the space ${\displaystyle X=\mathbb {R} ^{n}}$, the sequence of closed balls ${\displaystyle K_{i}=\{x:|x|\leq i\}}$ forms an exhaustion of the space by compact sets.

There is a weaker condition that drops the requirement that ${\displaystyle K_{i}}$ is in the interior of ${\displaystyle K_{i+1}}$, meaning the space is σ-compact (i.e., a countable union of compact subsets.)