Being open means that for every point in the set, there exists a neighborhood around that point that is entirely contained within the set.
A union of open balls is a set that consists of all open balls within a given set. An open ball is a set of all points within a certain distance from a specific point.
In order to prove that a set A is open, we need to show that for every point in the set, there exists an open ball around that point that is entirely contained within the set. This is essentially showing that A is a union of open balls.
Proving that a set A is open is important because it allows us to make conclusions about the continuity and differentiability of functions defined on that set. It also helps us understand the topological properties of the set.
One common technique is to use the definition of an open set and show that for every point in A, there exists an open ball around that point that is contained within A. Another technique is to use the properties of open balls, such as the fact that the union of open balls is an open set.