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Is a set which is an intersection of some connected set still connected? I think it is not true but could not think of an example.

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    intersection maybe?2012-12-05
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    Try to intersect two bananas.2012-12-05
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    Do you mean an intersection a collection of connected sets? Draw pictures in the plane.2012-12-05
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    @ArthurFischer The thing escaped me, but you are right, I definitely should have. :-)2012-12-05
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    @JasperLoy An excellent movie.2012-12-05
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    @did You should make your comment into an answer.2012-12-05
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    @MattN. Done. $ $2012-12-06
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    At least contained in [this](http://math.stackexchange.com/q/55646/8271)2013-02-26

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Try to intersect two bananas.$ $

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I think that is true. Take an example in $\mathbb{R}$, 3 intervals [0,1],[$\frac{1}{2}$,$\frac{3}{2}$],[$\frac{5}{4}$,2], their intersection is [$\frac{1}{2}$,1] and [$\frac{5}{4}$,$\frac{3}{2}$] is separated