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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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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 Image 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 Image is a real-valued function ...

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