fundamental group, free group

**mingcai6172** · March 21st 2009, 02:32 PM

Let $Y$ be the complement of the following subset of the plane $\mathbb{R}^2:$
$\{(x,0) \in \mathbb{R}^2: x \in \mathbb{Z} \}$
Prove that $\pi_1(Y)$ is a free group on a countable set of generators.

I don't know how to start this problem. Thanks in advance.

**aliceinwonderland** · March 22nd 2009, 07:41 PM

Originally Posted by mingcai6172

Let $Y$ be the complement of the following subset of the plane $\mathbb{R}^2:$
$\{(x,0) \in R^2: x \in \mathbb{Z} \}$
Prove that $\pi_1(Y)$ is a free group on a countable set of generators.

I don't know how to start this problem. Thanks in advance.

The deformation retract to the wedge product of circles would be the proper way to approach this problem.
Once you get the deformation retract, I think it would not be hard to prove this.

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