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The following is paraphrased from Lee's introduction to topological manifolds:

"A Fundamental fact about continuous functions is the extreme value theorem: A continuous real valued function on a closed bounded subset of $\mathbb{R}$ attains its maximum and minimum values.

The proof of this result hinges on the compactness of closed bounded subsets of $\mathbb{R}$.

This indicates that one might be able to formulate the extreme value theorem in more general situations, and it might be fruitful to study the notion of compactness further.further."

My guess is that there is no really simple way to motivate the modern definition compactness. Does anyone know when the modern definition appeared? Im guessing that the definition of sequential compactness appeared before the definition of compactness

Edit: The History behind the definition of compactness is given on the wikipedia page.

A Fundamental fact about continuous functions is the extreme value theorem: A continuous real valued function on a closed bounded subset of $\mathbb{R}$ attains its maximum and minimum values.

The proof of this result hinges on the compactness of closed bounded subsets of $\mathbb{R}$.

This indicates that one might be able to formulate the extreme value theorem in more general situations, and it might be fruitful to study the notion of compactness further.

My guess is that there is no really simple way to motivate the modern definition compactness. Does anyone know when the modern definition appeared? Im guessing that the definition of sequential compactness appeared before the definition of compactness

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A Fundamental fact about continuous functions is the extreme value theorem: A continuous real valued function on a closed bounded subset of $\mathbb{R}$ attains its maximum and minimum values.

The proof of this result hinges on the compactness of closed bounded subsets of $\mathbb{R}$.