## 5.1. Open and Closed Sets

### Examples 5.1.2(a):

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.