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.
A sequence can be defined as a function on the domain of natural number like \(1, 1/2, 1/3 ... 1/n\). This sequence approach to the 0, but never touch the 0. However, people can not take their desire to link the sequence and the 0. Because \(\infty\) is not a member of the natural number even real number, another concept is necessary to link the sequence and the 0. It is the limit.
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.