Limit Point

Definition

A limit point of a set is a point that can be approached by other points in the set, meaning that every neighborhood of the limit point contains at least one point from the set that is distinct from the limit point itself. This concept is crucial when discussing infinite sets, particularly in understanding how these sets can accumulate around certain points without necessarily including them.

5 Must Know Facts For Your Next Test

1. In an infinite set, a limit point can be essential for understanding how the set behaves, especially when considering sequences and convergence.
2. Not every infinite set has limit points; for example, the set of natural numbers does not have any limit points in the standard topology on the real numbers.
3. Limit points can be found within both bounded and unbounded sets, but their significance often lies in their relation to the completeness and compactness of sets.
4. The concept of limit points is directly related to Dedekind-infinite sets because these sets have properties that often rely on the existence of limit points for their analysis.
5. The collection of all limit points of a set forms another set known as the derived set, which plays an important role in topology.

Review Questions

• How do limit points differ from isolated points in the context of infinite sets?
  • Limit points are approached by other points in a set, meaning that there are other elements from the set arbitrarily close to them. In contrast, isolated points are those that do not have any other points from the set nearby; they stand alone without accumulation. Understanding this difference helps clarify how infinite sets can exhibit different behaviors based on whether they contain limit points or isolated points.
• Discuss how limit points relate to the concepts of closure and derived sets in the study of infinite sets.
  • Limit points play a crucial role in defining both closure and derived sets. The closure of a set includes all its limit points along with its original elements, ensuring it captures all points that can be approached by members of the original set. The derived set consists solely of limit points, which reveals how densely packed a set is within its space. These relationships highlight how limit points help us understand the structure and properties of infinite sets.
• Evaluate how recognizing limit points contributes to understanding Dedekind-infinite sets and their implications in topology.
  • Recognizing limit points is fundamental when evaluating Dedekind-infinite sets, as it allows us to see how these sets can possess infinite accumulation behavior. A Dedekind-infinite set is one that can be put into a one-to-one correspondence with a proper subset of itself. Limit points illustrate this property by showing how certain elements can perpetually be approached without ever being part of the original finite elements, thereby impacting our comprehension of convergence and continuity within topological spaces.

Related terms

Closure:

The closure of a set is the smallest closed set that contains the original set, including all its limit points.

Isolated Point: An isolated point of a set is a point in the set that does not have any other points of the set in its neighborhood, contrasting with limit points.

Dense Set: A dense set is one where every point in a space is either in the set or is a limit point of the set, indicating that it comes arbitrarily close to every point in that space.