Notes on Closed Sets

Introduction

• Two results to prove about closed sets:
  1. Union of a finite collection of closed sets is closed.
  2. Arbitrary intersection of closed sets is closed.

Preliminary Concepts

• De Morgan's Laws:
  • Important for understanding complements with unions and intersections.
  • Complement of a union = intersection of complements.
  • Complement of an intersection = union of complements.
• Definition of Closed Set:
  • A set is closed if its complement is open.
  • This definition connects closed sets to open sets using De Morgan's laws.

Results about Open Sets

• Arbitrary Union of Open Sets:
  • Open set.
• Finite Intersection of Open Sets:
  • Open set.
• Links to previously proven results on open sets.

Proofs

1. Finite Union of Closed Sets

• Given closed sets: U1, U2, ..., Un.
• Each closed set UK has an open complement.
• Consider the intersection of the finite collection of open complements:
  • Since this is a finite intersection of open sets, it is open.
• Taking the complement of this open intersection yields a closed set:
  • By De Morgan's laws, the complement of this intersection equals the finite union of the closed sets (U1 ∪ U2 ∪ ... ∪ Un).
• Therefore, the finite union of closed sets is closed.

2. Arbitrary Intersection of Closed Sets

• Consider closed sets U_alpha.
• Each closed set U_alpha has an open complement.
• The union of the open complements is open (by previous results).
• The complement of this arbitrary union is closed:
  • By De Morgan's laws, the complement of the union equals the intersection of the original closed sets (U_alpha).
• Thus, the arbitrary intersection of closed sets is closed.

Conclusion

• Summary of results proven:
  • Arbitrary union of open sets is open.
  • Arbitrary intersection of closed sets is closed.
  • Finite intersection of open sets is open.
  • Finite union of closed sets is closed.

Additional Thoughts

• Encouragement for questions in comments.