# Compactness

## Compactness

The compact is a property of space. In a nutshell, if space is compact, we can treat the space be a finite because space has a finite subcover. A continuous function on a compact space is uniformly continuous. Heine-Borel theorem describes the condition of compactness of finite dimensional space. Closed and bounded But the Heine-Borel theorem does not hold in an infinite-dimensional space. We need another condition. Previously, the compact space can be finite by taking subcover.

## Carathéodory's extension theorem

The studying sometimes starts with learning of boring preceding concepts. The highlight comes later. In history, the highlight concepts or the important problem were centered and the supporting concepts or lemmas followed. One of the central ideas of analysis is extension. The set of a rational number ($$\mathbb{Q}$$) extends to the real line $$\mathbb{R}$$. The Jordan measurable sets extend to the Lebesgue measurable sets ( $$\sigma -algebra$$ ). The outer measure can measure all subsets of $$X$$, whereas measure can only measure a $$\sigma -algebra$$ of measure set.