Connected Set
An open set \( A \) is considered connected if it cannot be divided into two disjoint (non-overlapping) open subsets \( A_1 \cap A_2 = \emptyset \) that are both non-empty \( A_1 \neq \emptyset \) and \( A_2 \neq \emptyset \), with their union forming the entire set \( A_1 \cup A_2 = A \).
In other words, there are no two open sets \( A_1 \) and \( A_2 \) such that:
- \( A_1 \) and \( A_2 \) are non-empty: \( A_1 \neq \emptyset \) and \( A_2 \neq \emptyset \)
- \( A_1 \) and \( A_2 \) are disjoint: \( A_1 \cap A_2 = \emptyset \)
- Their union equals the entire set \( A \): \( A_1 \cup A_2 = A \)
If even one of these conditions fails, the set \( A \) is not connected.
An Illustrative Example
Example 1: Connected Set
A classic example of a connected set is an open interval on the real number line, such as \( A = (0, 1) \).
This interval is connected because it’s impossible to split it into two non-empty, disjoint open subsets whose union would still cover \( (0, 1) \).
For example, if you attempt to divide \( (0, 1) \) into two non-empty, disjoint open subsets like \( (0, a) \) and \( (b, 1) \), where \( 0 < a < b < 1 \), their union would not encompass the entire interval \( (0, 1) \) because the interval \( (a, b) \) would be excluded. This means it fails to satisfy the condition \( A_1 \cup A_2 = A \).
Therefore, the set \( A = (0, 1) \) is indeed connected.
Example 2: Disconnected Set
An example of a disconnected set is the union of two separate intervals, such as \( A = (0, 0.4) \cup (0.6, 1) \).
This set is not connected because it can easily be divided into two non-empty, disjoint open subsets \( (0, 0.4) \) and \( (0.6, 1) \), whose union is still the original set.
The sets \( (0, 0.4) \) and \( (0.6, 1) \) are already open, disjoint, and non-empty. Their union exactly matches the given set, \( (0, 0.4) \cup (0.6, 1) \). Thus, the set \( A = (0, 0.4) \cup (0.6, 1) \) is not connected.
Alternative Definition of a Connected Set
A set \( A \) is connected if, for any pair of open sets \( A_1 \) and \( A_2 \) where:
- \( A_1 \cap A_2 = \emptyset \) (i.e., \( A_1 \) and \( A_2 \) are disjoint)
- \( A_1 \cup A_2 = A \) (i.e., the union of \( A_1 \) and \( A_2 \) covers the entire set \( A \))
then at least one of the sets must be empty.
If there exist two disjoint open sets that together cover the whole set \( A \), one of them must necessarily be empty.
This is because if both sets were non-empty, then \( A \) could be split into two separate parts, which contradicts the definition of a connected set.
And so on.
