Interactive Real Analysis5.1. 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 ?
Back
) and
[0, )
are open, closed, both, or neither ?
Back- 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.