# Interactive Real Analysis - part of MathCS.org

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## 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]
so that
• ( U S) ( V S ) = 0
• ( U S) ( V S ) = S
Note that the actual choice of U and V was not important. We could have chosen different intervals, e.g. (-50,0) and (0,10), to show both sets were disconnected. The important part was that the sets could be split at zero into two disjoint pieces.
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