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 ?
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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.

Interactive Real Analysis, ver. 1.9.5
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Page last modified: Mar 26, 2007