A bounded set is a collection of points in a space that is contained within a finite distance. In mathematical terms, a set is considered bounded if there exists a real number that can serve as a limit for the distances between points in the set. This means that you can draw a circle or sphere around the set, and all points will lie within that shape. For example, in the context of Euclidean space, a bounded set could be a closed interval on the number line, such as [0, 1]. This interval includes all numbers between 0 and 1, and it does not extend infinitely in either direction, making it a bounded set.