7. Connectedness

Connectedness

Definition and Properties of Connected Spaces

Connected Space: A topological space \( X \) is said to be connected if it cannot be divided into two disjoint nonempty open sets. In other words, there are no nontrivial clopen sets (sets that are both open and closed) in \( X \).

Properties of Connected Spaces:

• If \( X \) is connected, then every continuous function from \( X \) to the discrete two-point space \( \{0, 1\} \) is constant.
• If \( X \) is connected and \( A \) is a subset of \( X \), then the closure of \( A \) in \( X \) is connected.

Path-Connectedness and Its Implications

Path-Connected Space: A topological space \( X \) is said to be path-connected if for every pair of points \( x, y \) in \( X \), there exists a continuous function \( \gamma: [0, 1] \rightarrow X \) such that \( \gamma(0) = x \) and \( \gamma(1) = y \).

Implications of Path-Connectedness:

• Every path-connected space is connected, but the converse is not necessarily true.
• In a path-connected space, any two points can be joined by a path, implying a stronger form of connectedness.

Examples of Connected and Disconnected Spaces

Connected Space: The interval \( [0, 1] \) in the standard topology of \( \mathbb{R} \) is connected. However, the union of two disjoint intervals, such as \( [0, 1] \cup [2, 3] \), is disconnected.

Disconnected Space: The set of rational numbers \( \mathbb{Q} \) in the standard topology of \( \mathbb{R} \) is disconnected. This is because \( \mathbb{Q} \) can be partitioned into two disjoint open sets: the set of irrational numbers and the set of rational numbers.