Question

Prove or disprove: A set has no limit points if and only if each of its points is isolated.

Short answer

" Answer: Yes, a set has no limit points if and only if each of its points is isolated. This statement is proven by showing that if a set has no limit points, then each of its points is isolated, and conversely, if each point of a set is isolated, then the set has no limit points.

Step-by-step solution

Case 1: Proving that if a set has no limit points, then each of its points is isolated

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.

Case 2: Proving that if each point of a set is isolated, then the set has no limit points

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.

Key concepts

Isolated Points

In mathematical terms, an isolated point in a set is a point which stands alone without any closely neighboring points from the same set. This means there is a neighborhood around the isolated point that contains no other points from the set. For instance, consider a set with points scattered like islands in a sea: each point is an island by itself. To understand isolated points better:

• Every isolated point has a neighborhood around it that includes only the point itself and excludes other points of the set.
• They are crucial in examining the structure and properties of sets in real analysis.

Understanding isolated points helps determine the absence of limit points in a set, highlighting the independence of each point in that set.

Neighborhood

A neighborhood in real analysis refers to a range around a given point, within which the properties of the point are examined. Imagine a circle drawn around a point on a plane; this circle contains the point and possibly other points that lie within its circumference. In the context of isolated points and limit points:

• A neighborhood can be considered as an open interval or a small area around the point in question.
• For a point to be isolated, its neighborhood must contain no other points from the set.
• Conversely, for a point to be a limit point, every neighborhood of it must include at least one other point from the set.

This concept is key in distinguishing between isolated points and limit points, emphasizing the presence or absence of neighbors.

Real Analysis

Real analysis focuses on the behavior and properties of real numbers, sequences, and functions. A significant part of this field examines the limits and continuity within the number sets. In relation to isolated points and neighborhoods:

• Real analysis deals with the rigorous study of concepts like limits, helping to understand when a point is isolated or if it is part of a larger cluster of points.
• It involves examining how these points interact within different subsets of real numbers.
• This helps us conclude that a set has no limit points if and only if all its points are isolated.

Thus, real analysis is the framework from which the in-depth study of these concepts originates and supports conclusions like those found in the original exercise.