Definition:Compact Space/Motivation

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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.