Let A be a non empty part of ℝ, such that A and A complement are two open subsets of ℝ.
1- Prove that A is not bounded above.
2- Assuming that A complement is non-empty and let x in A complement.
and let B={xεA such that x≤t}. Prove that B is non empty and has a lower bound m such that m≥x.
3- Prove that m is not in A nor in A complement.
4- Let F be non-empty set in ℝ such that F and F complementary are closed in R. What can we say about the set F?
My attempt:
1-A is an open subset of R so we're going to use a proof by contradiction.
Suppose that a=sup(A) so for all x in A : x≤a . But since A is open then there exists an ε>0 such that:
]a-ε,a+ε[nA is nonempty so therefore there exist another number a+ε/2>a so therefore A is unbounded.
Thank you for any help before hand.