# Real Analysis/Topology problem

• Dec 8th 2013, 04:13 AM
Real Analysis/Topology problem
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.

• Dec 8th 2013, 05:18 AM
Plato
Re: Real Analysis/Topology problem
Quote:

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?

This is a trick question (or else whoever wrote it knows really little).
The only non-empty open subset of \$\displaystyle \mathbb{R}\$ having its complement open is \$\displaystyle \mathbb{R}\$ itself.

This easily seen by noting that \$\displaystyle \mathbb{R}\$ is a connected set.
• Dec 8th 2013, 05:30 AM