Motivation behind the definition of compact sets.

Asked Mar 19 '19 at 13:41

Active Mar 19 '19 at 13:41

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A subset $K$ of a metric space $X$ is said to be compact if every open cover of $K$ contains a finite subcover.

I want to know what's the motivation for constructing such sets?

I know, this definition leads to lots of interesting result, yet how people arrived at such construction?

And, why open cover gets the priority? I mean, particularly, why open sets are chosen?

asked Mar 19 '19 at 13:41

SOUL

@ YuiTo Cheng, I read the answers there,but I am not satisfied with anyone of it.Can you give a more clear picture? – SOUL Mar 19 '19 at 14:19
@Tom. Or What was the motivation behind the open covering definition of compactness – YuiTo Cheng Mar 19 '19 at 14:24

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