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This exercise was taken from the book "Fundamental Groups and Covering Spaces", from Elon Lages Lima.

"Let $X=C_1\cup\cdots\cup C_k$ be a finite union of convex open sets in the Euclidean space $\mathbb{R}^n$. Suppose that $C_i\cap C_j=\varnothing$ if, and only if, $j=i+1$ (or $j=i$, or $j=i-1$) or $\left\{i,j\right\}=\left\{1,k\right\}$. Prove that $X$ has the homotopy type of a circle."

The hint is to consider points $x_i\in C_i\cap C_{i+1}$ (and $x_0\in C_k\cap C_1$), and maybe try to deform $X$ to the polygon of sides $x_0x_1,\ldots,x_{k-1}x_0$.

Thank you.

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do you mean $C_i\cap C_j\neq\emptyset$ ? – Stefan Hamcke Jan 18 '13 at 22:10
If $k=3$ it can happen that $C_1\cap C_2\cap C_3\neq\emptyset$ and that $X$ is contractible. You have to require that for three different $C_i$ the common intersection is empty. Also, if $k=2$ then $X$ always deformation retracts to any point in the intersection. Did you forget some conditions from the exercise? – Stefan Hamcke Jan 18 '13 at 22:26

This is not exactly an answer to the question but something to help people. I'm trying to find this exercise on the Portuguese version of the book. Probably there is some error on the text.

On the English version, there are two exercises about this.

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On the Portuguese version, the same exercises.

enter image description here

So I believe that it is missing to require that any three of them have empty intersection.

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