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.

Definition: Connected and Disconnected

An open set S is called disconnected if there are two open, non-empty sets U and V such that:
1. U V = 0
2. U V = S
A set S (not necessarily open) is called disconnected if there are two open sets U and V such that
1. (U S) # 0 and (V S) # 0
2. (U S) (V S) = 0
3. (U S) (V S) = S
If S is not disconnected it is called connected.

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 seperated 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:

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:

Proposition: Connected Sets in R are Intervals

If S is any connected subset of R then S must be some interval.

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.

Definition: Totally Disconnected

A set S is called totally disconnected if for each distinct x, y S there exist disjoint open set U and V such that x U, y V, and (U S) (V S) = S.

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:

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.

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