# What is the interior of $S^n$

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.

• Interior as a subset of which topological space? May 10, 2013 at 5:46
• 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? May 10, 2013 at 5:49
• 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? May 10, 2013 at 5:53

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.