Closure of a Set

Definition

The closure of a set is the smallest closed set that contains the original set, which includes all its limit points. This concept is crucial in topology and algebraic geometry, as it helps to understand how sets behave in terms of convergence and limits, particularly within projective varieties. The closure of a set is often denoted as \(\overline{A}\) and reveals important properties about the underlying structure of geometric objects.

5 Must Know Facts For Your Next Test

1. The closure of a set can be constructed by taking the union of the set with its limit points, effectively adding those points that can be approached arbitrarily closely by points in the original set.
2. In projective geometry, the closure of a projective variety can provide insights into how varieties relate to each other, especially when studying intersections and unions.
3. The operation of taking closure is idempotent, meaning that taking the closure of a closed set will not change it.
4. For any subset \(A\) of a topological space, the closure \(\overline{A}\) is always a closed set and contains \(A\) itself.
5. In projective varieties, understanding closures can help in determining properties like dimension and irreducibility.

Review Questions

• How does the concept of closure relate to limit points within a given set?
  • The concept of closure directly involves limit points because the closure of a set includes all points in the original set along with its limit points. A limit point is defined such that every neighborhood around it contains at least one point from the original set. Therefore, when forming the closure, we ensure that not only are we considering the points explicitly in the original set but also any points that can be approximated by them.
• What are some implications of the closure operation in projective varieties regarding their properties?
  • In projective varieties, the closure operation helps determine critical properties such as dimension and intersection behavior. For example, when studying two intersecting varieties, their closures will contain all points relevant to their intersection which may not exist within their individual sets. This can reveal more about how these varieties interact and provide deeper insights into their geometric structure.
• Analyze how understanding closures can impact our approach to studying continuity and convergence in algebraic geometry.
  • Understanding closures allows us to analyze continuity and convergence in algebraic geometry more effectively by ensuring we account for all necessary limit points in our studies. For instance, when we define continuous functions between varieties, recognizing how closures interact ensures that we do not overlook behavior at boundary or limit points. This comprehensive view aids in building more robust arguments about convergence properties and continuity across different varieties.

Related terms

Closed Set: A closed set is a set that contains all its limit points, meaning that if a sequence in the set converges, its limit will also be included in 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, distinct from itself.

Topological Space: A topological space is a set equipped with a collection of open sets that satisfy certain axioms, forming the basis for concepts like continuity, convergence, and closure.