Is Every Subset of ℝ That Is Both Open and Closed Either ℝ or ∅?

[glebovg]

Homework Statement

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

Homework Equations

G ∩ bd(G) = ∅ ⇒ G is open

bd(F) ⊆ F ⇒ F is closed

bd(S) = bd(ℝ∖S) = bd(S')

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.

 

[SammyS]

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

 

[glebovg]

Isn't ℝ a clopen set?

 

[SammyS]

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