Let consider the three following sets :
$S_1 = \{ (x,y,z) \in \mathbb{R}^3\; |\; x^2+y^2+z^2=1 \}$
$S_2 = \{ (x,y) \in \mathbb{R}^2\; |\; \exists z\in \mathbb{R}, \; x^2+y^2+z^2=1 \}$
$S_3 = \{ x \in \mathbb{R}\; |\; \exists z\in \mathbb{R}, \; x^2+z^2=1 \}$
Am I right when saying : $S_1$ is of codimension 1 in $\mathbb{R}^3$,
$S_2$ is of codimension 1 in $\mathbb{R}^2$,
$S_3$ is of codimension 1 in $\mathbb{R}$
?