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Consider two $n$-connected sets $U,V\in \mathbb{R}^m$. What is the minimal $m$ such there exist $U,V$ such that $U\cap V$ is not connected? $n$-connected?

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I don't quite understand. Do you mean to ask "What is the smallest $m$ such that there are two $n$-connected subsets $U,V\subset\mathbb R^m$ whose intersection is disconnected/not $n$-connected?" You can have two spaces that are contractible, deformed balls for instance, whose intersection is not even connected. Think of two sets shaped like the letter U that intersect only at the extremities of the U. You get two path components this way. Am I misunderstanding your question? –  Olivier Bégassat Jul 20 '12 at 23:24
@OlivierBégassat Yes, you understood me. I'm still getting a hang for algebraic topology. –  pre-kidney Jul 20 '12 at 23:36
I think you need to specify the $n$ at here. For arbitrarily numbers this can be as bad as you wish. –  Bombyx mori Jul 21 '12 at 6:48

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