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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.