Closure of a set

Definition

The closure of a set refers to the smallest closed set that contains the original set, including all its limit points. This concept is essential in understanding how sets behave under continuous functions and helps in analyzing the properties of convergence and compactness in topology.

5 Must Know Facts For Your Next Test

1. The closure of a set can be obtained by taking the union of the set and its limit points.
2. In metric spaces, the closure of a set can also be defined as the intersection of all closed sets containing that set.
3. If a set is already closed, then its closure is the set itself.
4. Closure operations are idempotent; applying closure twice does not change the result after the first application.
5. The concept of closure is fundamental when dealing with continuous functions, as continuous images of closed sets are closed.

Review Questions

• How does the concept of closure relate to the properties of open and closed sets in topology?
  • Closure is directly related to both open and closed sets since it encompasses all limit points of a set, forming the smallest closed set containing it. Open sets consist of points that do not include their boundary points, while closed sets include all their limit points. Understanding closure helps distinguish between these two types of sets and illustrates how they interact with continuous functions.
• Discuss how the closure of a set can impact the continuity of functions defined on that set.
  • The closure of a set plays a crucial role in determining the continuity of functions. If a function is continuous on a set, then it remains continuous when extended to its closure. This is significant because it allows for analysis of limits and convergence behaviors at boundary points that are part of the closure but may not be included in the original set. This understanding aids in evaluating whether functions maintain their properties across boundaries.
• Evaluate how understanding the closure of a set can enhance one's ability to analyze convergence and compactness within topological spaces.
  • Understanding the closure of a set enhances analysis in topological spaces by providing insight into convergence and compactness. Since closure includes all limit points, one can identify convergence behaviors more effectively, as points approaching limits will be part of the closure. Additionally, compactness often relates to closed and bounded sets; recognizing how closure operates helps in determining whether sets meet criteria for compactness, thus allowing for deeper exploration into topological properties.

Related terms

Closed Set: A closed set is a set that contains all its limit points, meaning that any point that can be approached by points in the set is also included in the set.

Limit Point: A limit point of a set is a point such that every neighborhood of that point contains at least one point from the set different from itself.

Open Set: An open set is a set where, for every point within it, there exists a neighborhood around that point which is entirely contained within the set.