Infinite union of closed sets that isn't closed?

Click For Summary
SUMMARY

The discussion centers on identifying an infinite union of closed sets that is not closed. The example provided is the union of the closed intervals [0, x] for 0 ≤ x < 1, which results in the set [0, 1). This set is not closed because it does not include the endpoint 1. Additionally, the application of DeMorgan's laws to the intersection of open sets, such as (0, 1/n), is mentioned as another method to illustrate this concept.

PREREQUISITES
  • Understanding of closed sets in topology
  • Familiarity with the concept of unions and intersections of sets
  • Knowledge of DeMorgan's laws in set theory
  • Basic grasp of open and closed intervals
NEXT STEPS
  • Study the properties of closed sets in topology
  • Explore examples of unions of closed sets and their closure properties
  • Learn about DeMorgan's laws and their applications in set theory
  • Investigate the implications of open and closed intervals in real analysis
USEFUL FOR

Mathematicians, students of topology, and anyone interested in set theory and its applications in real analysis.

autre
Messages
116
Reaction score
0
So I have to find an infinite union of closed sets that isn't closed. I've thought of something that might work:

\bigcup[0,x] where 0\leq x&lt;1. Then, \bigcup[0,x] = [0,1), right?
 
Physics news on Phys.org
Yes, that is correct. 1 is not in that union because it is not in [0, x] for any x<1. If y< 1, however, there does exist x> y and y is in such [0, x]. Therefore the union contains all of [0, 1).
 
Or, you can apply DeMorgan to an intersection of opens that is not open, like (0,1/n).
 

Similar threads

Undergrad The set of nonzero values of pmf is at most countable
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad Construction of sigma-algebras: a counterexample
  • · Replies 5 ·
Replies
5
Views
3K
MHB Proof of a set union and intersection
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad How to determine if a set is a semiring or a ring?
  • · Replies 2 ·
Replies
2
Views
4K
Undergrad Intersection & union of closed and open sets?
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Countably Infinite Unions and the Real Numbers: Can They Really Be Uncountable?
  • · Replies 4 ·
Replies
4
Views
3K
Graduate Infinitary union combined with infinitary intersection
  • · Replies 2 ·
Replies
2
Views
4K
Undergrad Countability of binary Strings
  • · Replies 18 ·
Replies
18
Views
4K
MHB Axioms of Set Theory .... and the Union of Two Sets ....
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad How does a disjoint union differ from a set of sets?
  • · Replies 4 ·
Replies
4
Views
2K