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I need to calculate the fundamental groups of the following spaces:

$X_1 = \{ (x, y, z) \in \mathbb{R}^3 | x \neq 0\} $

$X_2 = \mathbb{R}^3 \backslash \{ (x, y, z) | x = 0, y = 0, 0 \leq z \leq 1 \}$

$X_3 = \mathbb{R}^3 \backslash \{ (x, y, z) | x= 0, 0 \leq y \leq 1 \} $

I'm not sure at all how one would calculate these. I think that $X_1$ is still a convex space, so the fundamental group might be {1} but I'm really not at all sure...I need to calculate the fundamental groups of the following spaces:

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    The fundamental group of X2=X1, and the first groups is the same of $\mathbb{S}^1$.2012-03-17
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    Your first space is not path-connected, it doesn't make sense to speak of "the" fundamental group for it (even if the fundamental groups for any base point is the same in this case).2012-03-17
  • 0
    Contrary to chessmath's comment, $X_1$ is not $\mathbb{S}^1$, and because the first isn't even connected I don't like saying it has the same fundamental group as $X_2$ (and $X_2$ is, incidentally, deformation retractable to a sphere).2012-03-17

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