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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.

Now the weird open cover we had no longer covers the whole interval because the points 0 and 1 aren’t any of the intervals.

It’s harder to show that we couldn’t cook up a different pathological open cover, so you’ll have to take my word for it for now.

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.