What is the one point compactification of $S^n\times \mathbb{R}$?
What is the one point compactification of $S^n\times\mathbb{R}$?
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general-topology
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3$S^n\times S^1/S^n\times\{\text{pt}\}$ – 2011-04-06
1 Answers
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It might be helpful to think of this as $\mathbb{S}^n \times [0,1]$ with the boundary identified to a point. It's easy to see that $\mathbb{S}^n \times \mathbb{R}$ embeds homeomorphically as an open dense set by the product of the identity on $\mathbb{S}^n$ with any homeomorphism from $\mathbb{R}$ to $(0,1)$. Since the complement of the image of $\mathbb{S}^n \times \mathbb{R}$ is a singleton, this is a one-point compactification of $\mathbb{S}^n \times \mathbb{R}$.
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0Wait, I think I can see it now. The "top half" corresponds to the upper hemisphere, while the "bottom half" corresponds to the lower hemisphere. Makes sense. – 2016-09-16