I need some clarification about one-point compactifications. In one of my exercise I want to show what is the space $X$ such as its compactification is $S^3\setminus\gamma$ where gamma is a circumference of $S^3$.
With my little knowledge of compactification I know that the compactification of any $\mathbb{R}^n$ is $S^n$, so I suppose that the space $X$ is $\mathbb{R}^3\setminus\delta$ where $\delta$ is a straight line in $\mathbb{R}^3$.
How can I do that?
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Barbamento
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Probably not the answer you're looking for but the one point compactification of a compact set is itself. – andrew Apr 09 '21 at 13:34
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Well the exercise specifically says to use the fact that $S^3$ is the compactification of $\mathbb{R}^3$ so I'm kinda forced that way – Barbamento Apr 09 '21 at 13:41
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2Well, if I understand $\gamma$ correctly, then $S^3\setminus\gamma$ is not compact, hence can't be the compactification of anything. – Berci Apr 09 '21 at 15:36