Closed set

Definition

A closed set is a fundamental concept in topology, defined as a set that contains all its limit points. In a topological space, a set is closed if its complement is open, meaning that for any point not in the set, there exists an open neighborhood around that point that does not intersect with the closed set. This concept is key to understanding continuity and convergence within spaces and relates closely to other properties of topological spaces.

5 Must Know Facts For Your Next Test

1. In any topological space, the intersection of any collection of closed sets is also a closed set.
2. The union of any finite number of closed sets is closed, while the union of infinitely many closed sets may not be closed.
3. A closed interval [a, b] in the real numbers is an example of a closed set, as it contains all its limit points.
4. In Hausdorff spaces, closed sets can be used to separate points, as any two distinct points can be contained in disjoint closed neighborhoods.
5. Every compact subset of a Hausdorff space is closed, which has significant implications in analysis and topology.

Review Questions

• How do closed sets relate to open sets in a topological space?
  • Closed sets and open sets are complementary concepts in topology. A set is defined as closed if its complement in the space is an open set. This relationship helps establish the structure of the topology and plays a crucial role in understanding continuity and convergence. For example, if we know a set is closed, we can infer properties about its boundary and limit points based on its complement.
• What role do limit points play in defining closed sets, particularly in relation to continuous functions?
  • Limit points are essential to defining closed sets because a set is considered closed if it contains all its limit points. This means that if you have a sequence converging to a point within the set, that point must also be included in the set for it to be classified as closed. In terms of continuous functions, if the preimage of an open set under a continuous function is open, then the preimage of a closed set is also closed. This connection underscores the importance of closed sets in analyzing continuity.
• Evaluate how the properties of closed sets influence convergence in Hausdorff spaces and provide examples.
  • In Hausdorff spaces, one of their defining properties is that for any two distinct points, there exist disjoint neighborhoods around each point. This property greatly influences convergence because it guarantees that limits of sequences are unique. For example, if a sequence converges to two distinct points in a Hausdorff space, this would violate the property of convergence since there cannot be neighborhoods that separate these points from others in their respective limits. Therefore, closed sets serve as useful tools for establishing limits and ensuring separation of points within these spaces.

Related terms

Open Set: An open set is a set that, for every point in the set, there exists an open neighborhood completely contained within the set.

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

Closure: The closure of a set is the smallest closed set that contains the original set, encompassing all limit points of that set.