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I'm trying to determine the significance of a compact subset of a metric space in relation to calculus in general.

I know the definition: A subset [itex]\small K[/itex] of a metric space [itex]\small X[/itex] is said to be compact if every open cover of [itex]\small K[/itex] contains a finite subcover.

But what is the importance ofeveryopen cover containing a finite subcover? For example suppose I have a subset [itex]\small E[/itex] of a metrics space [itex]\small X[/itex], which is open relative to [itex]\small X[/itex]. I assign to each [itex]p \in E[/itex] a positive number [itex]r_{p}[/itex] such that the conditions [itex] d(p,q) < r_{p}[/itex], [itex]q \in X[/itex] imply [itex]q \in E[/itex]. This is an open cover of [itex]\small E[/itex], which does not have a finite subcover which contains [itex]\small E[/itex], correct? If this is true, then I'm thinking this must be why no open set in [itex]\small X[/itex] can be compact, because we can find an open cover with no finite subcover. (the one above)

Additionally, can't we obtain an open cover of this same [itex]\small E[/itex] with a finite subcover if we simply pick neighborhoods of a finite set of points in [itex]\small E[/itex] with radius large enough that [itex]\small E[/itex] is contained in the union of these neighborhoods? Hence the importance ofevery, as noted before

If this is all true, then my point is I don't see the 'big picture' importance of these observations as they relate to the development of calculus? What are the implications?

Furthermore, I know things such as every sequence in a compact metric space [itex]\small X[/itex] has a subsequence which converges to a point in [itex]\small X[/itex], and the theorems about all Cauchy sequences converging if in a compact metric space.

I'm struggling to tie things together...

As always, thanks

pob

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# What is the significance of compactness?

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