### III.9 Compactness and Compactification

*Terence Tao*

In mathematics, it is well-known that the behavior of finite sets and the behavior of infinite sets can be rather different. For instance, each of the following statements is easily seen to be true whenever *X* is a finite set but false whenever *X* is an infinite set.

**All functions are bounded.** If *f : **X* → is a real-valued function on *X*, then *f* must be bounded (i.e., there exists a finite number *M* such that |f(x)| *≤* M for all x ∈ X).

**All functions attain a maximum.** If *f : **X* → is a real-valued function ...