Proof That Every Open Cover of a Closed and Bounded Interval is Reducible to a Finite Subcover


Every open cover of a closed and bounded intervalis reducible to a finite subcover.


Ifis a closed and bounded interval with an open coverso that

Now apply the Heine - Borel theorem which states that a subspace of(with the usual topology) is compact if and only if it is closed and bounded. Sinceis closed and bounded it is compact hence has a finite subcover ofso that

for some

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