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5. Topology
5.3. Connected and Disconnected Sets
In the last two section we have classified the open sets, and looked at two classes of
closed set: the compact and the perfect sets. In this section we will introduce two other
classes of sets: connected and disconnected sets.
Note that the definition of disconnected set is easier for an open set
S. In
principle, however, the idea is the same: If a set
S can be separated into two
open, disjoint sets in such a way that neither set is empty and both sets combined give the
original set
S, then
S is called disconnected.
To show that a set is disconnected is generally easier than showing connectedness: if you
can find a point that is not in the set S, then that point can often be used to
'disconnect' your set into two new open sets with the above properties.
Examples 5.3.2: 


Is the set { x R :  x  < 1, x # 0 }
connected or disconnected ? What about the set
{ x R :  x 
1, x # 0 }

Is the set [1, 1] connected or disconnected ?

Is the set of rational numbers connected or disconnected ? How about the irrationals ?

Is the Cantor set connected or disconnected ?

In the real line connected set have a particularly nice description:
Hence, as with open and closed sets, one of these two groups of sets are easy:
 open sets in R are the union of disjoint open intervals
 connected sets in R are intervals
The other group is the complicated one:
 closed sets are more difficult than open sets (e.g. Cantor set)
 disconnected sets are more difficult than connected ones (e.g. Cantor set)
In fact, a set can be disconnected at every point.
Intuitively, totally disconnected means that a set can be be broken up into two pieces at
each of its points, and the breakpoint is always 'in between' the original set.
Example 5.3.5: 


The Cantor set is disconnected. Is it totally disconnected ?

Is the set {0, 1} connected or disconnected ? Is it totally disconnected ?

Is the set {1, 1/2, 1/3, 1/4, ...} totally disconnected ? How about the set
{1, 1/2, 1/3, 1/4 ...} {0} ?

Find a totally disconnected subset of the interval [0, 1] of length 0 (different
from the Cantor set), and another one of length 1.
