Give an example of an open cover of the set 68 that has no f

Give an example of an open cover of the set (6,8) that has no finite subcover.

Solution

The answer is the sets (1/n,1) for n1.

Explanation:

For n>0n>0 let an=1nan=1n. Now consider the intervals (an+1,an)(an+1,an). Their union covers the set (0,1){an/nN⁺}. Let I_n be an interval which covers anan and is small enough.

Clearly {I_n}_nN⁺{(a_n+1,a_n)}nN+{In}_nN⁺ is an open cover of (0,1)(0,1). Argue that it is impossible to have a finite subcover.

If such finite subcover would exist then it would only contain a finite number of intervals of the form (a_n+1,a_n). This means that for some large enough NN we have that (0,a_N) is contained completely in a finite number of I_n\'s. Argue that (0,a_N) cannot be contained in _K=N^I_k, because they are so small (i.e. their sum will never aggregate to 1/N) and derive a contradiction.