51�Թ�

The real definition of compactness is that a space is compact if every open cover of the space has a finite subcover.

So the number line is not compact because we have found an open cover that does not have a finite subcover.

That’s the point of the finite subcover in the definition of compactness.

Proving noncompactness only requires producing one counterexample, while proving compactness requires showing that every single open cover of a space, no matter how oddly constructed, has a finite subcover.

If you’ve taken a topology class before, you might have seen the definition of the topological property called compactness: a set is compact if every open cover of the set has a finite subcover. The topologist’s sine curve is not compact, but the closed topologist’s sine curve is.