# Clopen Set

1. Sep 30, 2011

### glebovg

1. The problem statement, all variables and given/known data

Show that if A ⊆ ℝ is both open and closed then A is either ℝ or ∅.

2. Relevant equations

G ∩ bd(G) = ∅ ⇒ G is open

bd(F) ⊆ F ⇒ F is closed

bd(S) = bd(ℝ∖S) = bd(S')

3. The attempt at a solution

Suppose A is a clopen set such that it is neither ℝ nor ∅ then ℝ∖A = A' is neither ℝ nor ∅. Now, A' is open because A is open (and closed). So ℝ = A ∪ A', where both A and A' are closed, which implies that ℝ is closed, a contradiction.

2. Sep 30, 2011

### SammyS

Staff Emeritus
Why is the fact that "ℝ is closed" a contradiction. ℝ is indeed closed.

3. Sep 30, 2011

### glebovg

Isn't ℝ a clopen set?

4. Sep 30, 2011

### SammyS

Staff Emeritus
What's the definition of a "clopen" set?