Let and . Show that is an open cover of .

So choose . Pick such that . This implies that . Is this correct? Could I pick in a different way?

What if we had a finite subset ?

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- Jul 11th 2008, 11:46 AMparticlejohncovering
Let and . Show that is an open cover of .

So choose . Pick such that . This implies that . Is this correct? Could I pick in a different way?

What if we had a finite subset ? - Jul 11th 2008, 12:26 PMPlato
If you had a finite subset of the collection, then the smallest left-hand endpoint would a number .

But and .

Therefore, the subcollection does not cover . - Jul 11th 2008, 02:01 PMPlato
The statement that set is compact means

**every open covering has a finite subcover**.

Having one open cover which has a finite subcover does not imply that the set is compact.

Consider:

That is clearly an open covering of and the collection contains a finite subcover.

But is still not compact.