Codimension of a sphere

Question

Let $\cal{S}$ be the n-sphere as a n-dimensionnal submanifold of $R^{n+1}$.

The codimension of $\cal{S}$ in $R^{n+1}$ is $1$ right ?

Answer 1

Right. In general, if $M$ is an $n$-manifold and $S\subseteq M$ is a $k$-dimensional submanifold, then the codimension of $S$ in $M$ is $n-k$ by definition.