## 5.2. Compact and Perfect Sets

### Examples 5.2.7(b):

Can you find infinitely many closed sets such that their
intersection is empty and such that each set is contained in
its predecessor ? That is, can you find sets

It is easy to simply find some closed sets with empty intersection.
For example, the intersection of all intervals of the form
*A*such that_{j}*A*and_{j+1}A_{j}*A*?_{j}= 0*[n, n+1]*is certainly empty.

To find sets contained in one another is slightly more complicated.
We might try sets of the form
*A _{j} = [0, 1 / j]* for all

*j*. Then

*A*, but their intersection contains the point {0}.

_{j+1}A_{j}
Let
*A _{j} = [j, )*.
Then

*A*, because

_{j+1}A_{j}*[j + 1, ) [j, )*. But by the Archimedian principle, the intersection of all sets

*A*is empty.

_{j}