91Ó°ÊÓ

Let A be the set with no limit points. We want to show that each point x in A is isolated. We can do this by finding a neighborhood around x that only contains x and no other points from A. Since x is not a limit point, there must exist a neighborhood, N, around x such that N contains no points from A different from x itself. Thus, x is an isolated point of A. This is true for all points in A, so if A has no limit points, each of its points is isolated.

Let A be a set where each point is isolated. We want to show that A has no limit points. To do this, we assume that there exists a point x that is a limit point of A and then demonstrate a contradiction. Since x is a limit point, we know that every neighborhood of x must contain at least one point of A different from x itself. However, we were given that each point of A is isolated, which means there must exist a neighborhood of x, N, that contains no points of A other than x itself. This contradicts our assumption that x is a limit point. Therefore, if each point of a set is isolated, the set has no limit points. Since we have proven both cases, we can conclude that a set has no limit points if and only if each of its points is isolated.