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Let $(E,d)$ a metric space. We say that $E$ is a connected space if the only subsets which are both open and closed (clopen sets) are $E$ and the empty set. A subset of $E$ is connected if is a connected subspace of $E$. Let $\{C_i\}_{i\in I}$ a family of connected subsets of $E$. Is $$\bigcap_{i\in I} C_i$$ a connected subset of $E$?

Thanks for any boost.

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Try intersecting a circle and a line in $\mathbb{R}^2$. – Pete L. Clark Aug 4 '11 at 18:52
Nice! :) :) :) :) – leo Aug 4 '11 at 19:01
up vote 8 down vote accepted

[Sorry, I should know by now to leave answers as answers, not as comments.]

Once more:

Try intersecting a circle and a line in $\mathbb{R}^2$.

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