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Homework Statement
Let ε = { (-∞,a] : a∈ℝ } be the collection of all intervals of the form (-∞,a] = {x∈ℝ : x≤a} for some a∈ℝ.
Is ε closed under countable unions?
Homework Equations
Potentially De Morgan's laws?
The Attempt at a Solution
Hi everyone,
Thanks in advance for looking at my question. I know intuitively this IS closed under countable unions; the problem is showing this rigorously.
Here is what I have done so far:
ε is closed under finite unions.
Proof:
Use induction:
n=2 case:
A1 = (-∞,a1], A2 = (-∞,a2], a1<a2 (trivial if equal).
Then A1 ∪ A2 = A2 ∈ ε
Assume true for n=k.
i.e. A1 ∪...∪ Ak = (-∞,am], where am = {max(ai : i = [1,k]}
For n=k+1:
A1 ∪...∪ Ak ∪ Ak+1 = (-∞,am] ∪ (-∞,ak+1], which is in ε by the k=2 case.
How do I then extend this argument to the infinite case?
Also, using a similar induction argument, ε is closed under finite intersections.
ε is NOT closed under complements e.g. if A = (-∞,a], then Ac = (a,∞) ∉ ε. As it is not closed under complements, I don't think the answer can involve De Morgan's laws.
Thanks for any help!