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## Main Question or Discussion Point

I've only just started getting into Topology and a few examples of compactedness have me a little confused. For instace, the one in the title: how is the open interval (0,1) not compact but [0,1] is? Obivously I'm making some sort of logical mistake but the way I think about it is that there are any number of open covers for (0,1). I'll use {(-2,2),(-1,3),(0,4)}. By definition, there needs to exist a finite subcover which still contains (0,1) right? So couldn't that finite subcover simply be {(-1,3)}? I have a feeling the reason I'm not grasping this example is because the definition states that a finite subcover must exist for all open covers of a given set, but I still can't think of an open cover which does not have a finite subcover which contains (0,1). Thanks.