(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# Homework Help: Clopen Set

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