As the topic, how to prove that the only set in $\mathbb{R^1}$ which are both open and close are the $\mathbb{R^1}$ and $\emptyset$. I tried to prove by contradiction, but i can't really show that the assumption implies the contrary.
Are the only sets in $\mathbb{R^1}$ which are both open and closed $\mathbb{R^1}$ and $\emptyset$?
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general-topology
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1Suppose that $X$ is both open and closed and that $a\in X$. Let $S = \lbrace x\ge a: [a,x]\subset X\rbrace$. Is $S$ bounded above? – 2012-09-23
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0not sure if S is bounded or not – 2012-09-23