## 5.1. Open and Closed Sets

### Examples 5.1.2(c):

Are the sets
{1, 1/2, 1/3, 1/4, 1/5, ...} and
{1, 1/2, 1/3, 1/4, ...} {0}
open, closed, both, or neither ?

- The set {1, 1/2, 1/3, 1/4, 1/5, ... } is not open, because it
does not contain any neighborhood of the point
*x = 1*. - The complement of the set {1, 1/2, 1/3, 1/4, 1/5, ... }
contains the number 0. But if
*(-a, a)*is any neighborhood of 0, then there exists an*N*so large such that*1/N < a*. This neighborhood is not part of the complement, because it contains the element*1/N*from the set. Therefore the complement is not open. That means, however, that the original set is not closed. - The set
*{1, 1/2, 1/3, 1/4, 1/5, ... } {0}*is not open because it does not contain any neighborhood of the point*x = 1*. - For the last question, we need to look at the complement of
the set
*{1, 1/2, 1/3, 1/4, 1/5, ... } {0}*:*comp( {1, 1/2, 1/3, 1/4, 1/5, ... } {0} ) = (1, ) (-, 0)*