# Prove that the spaces have the same homotopy type

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