What is connected sets?

A connected set is a set where every pair of points can be connected by a path within the set. This means that all points in the set are "reachable" in a sense.

For example, a line segment is a connected set because any two points on the segment can be connected by a straight path on the segment. However, a disjoint set of two line segments is not connected because there is no path that connects the two segments.

Connected sets are important in topology and geometry because they allow us to study the properties of a space as a whole rather than just looking at individual points. For example, we can talk about the connectedness of a graph, which tells us how easy it is to travel from one node to another.

It is also useful to note that the complement of a connected set may not be connected. For example, the complement of a closed disk is two disjoint open sets.