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I have solved a problem from Hatcher's book which can be found here Fundamental group of $\mathbb{R}^3$ \ finite number of lines passing through origin.

So, motivated by this problem, I was thinking about problem like computation of fundamental group of $\Bbb R^3-\{ax+by+cz=0\}$ And generalization of this problem as like I mentioned above.

Can you please give me any idea how to solve this problem?

Thanks in advance.

Bernard
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Rodriguez J.
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    Once you fix a base point you get one of the two connected components which is homotopically equivalent to $\mathbb{R}^3$ and hence is trivial. – Quimey May 13 '21 at 09:39
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    $\mathbb R^3$ minus a plane is just two disjoint open half-spaces, both contractible. – lisyarus May 13 '21 at 09:40
  • @HennoBrandsma Ah yes, you're right. I'm not thinking straight. – Teddy38 May 13 '21 at 12:14

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$\Bbb R^3 \setminus P$ where $P$ is a plane is just a disjoint sum of two copies of $\Bbb R^3$. If $p$ is the base point for the fundamental group, it's in one of these copies so the fundamental group is just the one for that copy (as a loop only lies inside that copy for connectedness reasons) and thus trival (as the space is contractible), i.e. $\{0\}$.

Henno Brandsma
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