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I have two subsets of $\mathbb{R}^3$, $X$ and $Y$ defined as $X=\{(x,y,t) \mid x^2+y^2=1,t \in \mathbb{R} \}$ and $Y= \{(0,e,t) \mid e^2=1,t \in \mathbb{R} \}$. Are $X$ and $Y$ homeomorphic? Are $\mathbb{R}^3 \backslash X$ and $\mathbb{R}^3 \backslash Y$ homeomorphic?

For the first one, i think the fundamental groups are not same so they can't be homeomorphic.

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$X$ is just an infinite hollow cylinder, $S^1\times\Bbb R$, and $Y$ is two parallel lines in $\Bbb R^3$; if you can see that, you should have no trouble following Seirios’s answer. – Brian M. Scott Jan 28 '13 at 20:20

$X$ and $\mathbb{R}^3 \backslash Y$ are connected whereas $Y$ and $\mathbb{R}^3 \backslash X$ are not connected.

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