Examples 5.3.2(a):
Is the set { x R : | x | < 1,
x # 0 } connected or disconnected ? What about the set
{ x R : | x |
1, x # 0 }
The first set is open, so we can try to use the easier definition of disconnected.
It is indeed disconnected; it can be 'split at 0'. More precisely, let
U = (-1,0) and V = (0,1). Then U and V have empty
intersection and their union is the original set. But that means the original set
must be disconnected.
The second set is also disconnected, but since it is not open we must use the second part of the definition of disconnected sets. Again, we can 'split' the set at zero. Let U = (-2,0) and V = (0, 2) and set S = [-1, 0) (0, -1]. Then both sets U and V are open and we have that
- U S = [-1, 0)
- V S = (0, 1]
- ( U S) ( V S ) = 0
- ( U S) ( V S ) = S