The connected set closure is a mathematical concept that refers to the smallest closed set that contains a given connected set. It is also known as the closure of a connected set or the topological closure.
A set is considered disconnected if it can be divided into two non-empty subsets that are separated from each other by a gap or a hole. In other words, there is no continuous path that connects the two subsets.
To show that a set is not disconnected, you can prove that there is no gap or hole that separates the set into two non-empty subsets. This can be done by finding a continuous path that connects any two points in the set, therefore showing that the set is connected.
The connected set closure is used to determine whether a set is disconnected. If a set is not disconnected, then its connected set closure will be the same as the set itself. However, if a set is disconnected, its connected set closure will be larger than the set, as it includes all the points of the set plus the points that bridge the gap between the disconnected subsets.
Sure, let's take the set of real numbers between 0 and 1, denoted as [0,1]. To show that this set is not disconnected, we can prove that there is a continuous path that connects any two points in the set. For example, we can draw a straight line from 0 to 1, passing through all the points in between, thus showing that the set is connected. Therefore, the connected set closure of [0,1] would be the same as the set itself.