5.1. Open and Closed Sets
Which of the intervals (-3, 3), [4, 7], (-4, 5], (0, ) and [0, ) are open, closed, both, or neither ?
- The interval (-3, 3) is open, because if x is any number in (-3, 3), then -3 < x < 3. or equivalently, -3 - x < 0 < 3 - x. Now let = min( 3 + x, 3 - x ). Then > 0, and the interval (x - / 2, x + / 2) is contained in (-3, 3). In fact, the same argument works for any interval of the form (a, b) with a, b real numbers.
- The interval [4, 7] is closed, because its complement consists of the two open sets (- , 4) and (7, ).
- The interval (-4, 5] is neither open nor closed. It is not open, because the point x = 5 is contained in the set, but every neighborhood of that point is not contained inside the set. It is not closed, because its complement (- , -4] and (5, ) is not open at x = -4.
- The interval (0, ) and also the interval (- , 0) are both open, because every point in the set contains a neighborhood contained inside the set.
- The interval [0, ) and also the interval (- ,0] are both closed, because their complements are the open sets mentioned above.