Definition:Compact Space/Motivation

Motivation for the Concept of Compact Space
The question is asked:
 * What is compactness useful for?

to which the following answer can be presented:
 * $(1): \quad$ Local properties can be extended to being global properties.


 * $(2): \quad$ Compactness allows us to establish properties about a mapping, in particular continuity, in a context of finiteness.

In particular it can be noted that many statements about a mapping $f : A \to B$ are:


 * $\text {(a)}: \quad$ Trivially true when $f$ is a finite set


 * $\text {(b)}: \quad$ true when $f$ is a continuous mapping when $A$ is a compact space


 * $\text {(c)}: \quad$ false, or very difficult to prove when $f$ is a continuous mapping but when $A$ is not compact.