If $V = [0,1] \times [0,1] \subset \mathbb{R}^2$. We define the equivalence relation $\sim$ on $V$ as follows: every element $(x,y) \in V$ is equivalent with itself and besides that the three elements $(0,x), (1,x)$ and $(1-x,0)$ are equivalent for $x \in [0,1].$ We note $X$ as the quotient space belonging to $p: V \rightarrow X$ for the canonical quotient map. Now define $U_1 :=V \backslash \{(1/2,1/2) \}$, and $U_2 :=(0,1)\times(0,1)$ and note $W_i :=p(U_i)$ for $i =1,2$.If $x\in W_1 \cap W_2$.

Prove that $\pi_1(W_1,x) \simeq \mathbb{Z}*\mathbb{Z}$.

If we want to do this we need to know what $W_1$ looks like but I can't wrap my head around it.. if $(0,x)$ and $(1,x)$ we get a cylinder but now we also have $(1-x,0)$. Could anyone show me how I could visualize this or other quotient spaces better? Thanks.

Stefan Hamcke
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  • The three points that are equivalent: Is that for all $x \in [0,1]$? – N. Owad Jul 02 '15 at 13:31
  • The [dunce hat](http://course1.winona.edu/eerrthum/math450/DunceHat.jpg) is not exactly the same space, but may be close enough to help you visualize. The wikipedia [animation](https://en.wikipedia.org/wiki/Dunce_hat_%28topology%29) may be helpful as well. – Andrew D. Hwang Jul 02 '15 at 13:48

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