# 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.

## Sources

- 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