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.

1960: The role of compactness in analysis (Amer. Math. Monthly Vol. 67: pp. 499 – 516) www.jstor.org/stable/2309166
1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $5$: Compact spaces: $5.2$: Definition of compactness