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I have the following question:

"Is every set in the Euclidean Topology either open or closed? (exclusive or)"

I would say that the answer is FALSE, because for example I can have the interval (1,2], which is open at 1 and closed at 2.

Thanks, Daniel.

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Should be false because the entire set of $\mathbb{R}$ is both open and closed. – Q.matin Feb 18 '13 at 9:58
Your example is fine: it is a subset of $\Bbb R$ that is neither open nor closed. Others have pointed out the other kind of counterexample, a set that is both open and closed; in $\Bbb R$ there are only two, $\varnothing$ and $\Bbb R$. – Brian M. Scott Feb 18 '13 at 9:59
See this link for the completing answer. – Babak S. Feb 18 '13 at 10:08
Beware though that the example you provided: $(1,2]$, is not both open and closed, but neither closed nor open, despite having one end 'open' and the other end 'closed'. Think about the definitions of open for this to make sense. – user54147 Feb 18 '13 at 10:27
Thanks, @LevLivnev. Yes, I had to ponder at the problem for a sec. Thank you all for such a complex answer. – Daniel Feb 18 '13 at 10:34
up vote 2 down vote accepted

It is false, because $\mathbb{R}^n\subset \mathbb{R}^n$ is always open and closed, and $\varnothing$ is clopen too.

In fact in every topological space there are at least 2 clopen sets, all the space, and the empty set. In addition a topological space $T$ is connected iff the only clopen sets are $T$ and $\varnothing$

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In fact this answer (with trivial changes) apply to all topologies, not just the Euclidean one. – Willie Wong Feb 18 '13 at 9:54
i will add this thanks – Dominic Michaelis Feb 18 '13 at 9:55
I understand. Thank You. – Daniel Feb 18 '13 at 9:56

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