Example:
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} ) =
(prove it !)
- This set is the union of open sets, hence it is open.
Therefore the original set is closed.
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