Is the interior of $S^n$ empty or is it all of $S^n$? My reasoning is that given any point on $S^n$, any open ball around the point cannot be contained in $S^n$, so it should be empty. However, it's all of $S^n$ according to my book.
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8$\begingroup$ Interior as a subset of which topological space? $\endgroup$– Jonas MeyerMay 10, 2013 at 5:46
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$\begingroup$ Well $S^n \subset \mathbb{R}^{n+1}$, right? Simply applying the definition of interior point, it seems to me no point of $S^n$ can be an interior point. Where am I going wrong? $\endgroup$– user77134May 10, 2013 at 5:49
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5$\begingroup$ It is true that you could consider $S^n$ as a subspace of $\mathbb R^{n+1}$, but it need not be thought of as such. The $n$-sphere is a topological space in its own right. To talk about the interior of a set, you have to know in which topological space you are working. In the one-point topological space $\{0\}$, the interior of $\{0\}$ is $\{0\}$, but as a subspace of $[0,1]$ with its standard topology, the interior of $\{0\}$ is empty. So, can you find further context about what your book is saying? The interior of $S^n$ as a subset of which topological space? $\endgroup$– Jonas MeyerMay 10, 2013 at 5:53
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To emphasize what is implicit in previous comments and answers: It makes no sense to speak of the interior of a set $A$ unless one also specifies a topological space $X$ with $A\subseteq X$. So we should speak of the interior of $A$ in $X$. This interior depends on both $A$ and $X$, so if you change $X$ while leaving $A$ unchanged, you may very well find a changed interior.
answered May 10, 2013 at 15:57