I am given the following subset $S$ of $\mathbb{R}^{10}$,
\begin{cases} |z_1|^2+|z_2|^2+|z_3|^2+|z_4|^2+|z_5|^2=1 \\ z_1^3+z_2^5+z_3^2+z_4^2+z_5^2=0 \end{cases}
and asked to show that it is a regular smooth submanifold of $\mathbb{R}^{10}$. (This is simply some part of $\mathbb{C}^5$ identified with $\mathbb{R}$ in the natural way.)
I understand that this amounts to showing that the space is second countable, Hausdorff and exhibits a maximal atlas.
I think that it must be Hausdorff since it is a subspace of $\mathbb{R}$. However, with regards to showing second countability and exhibiting a maximal atlas, I am completely lost. I think the dimension must be $\leq 9$ since the first constraint on the elements of the subspace can be identified with a surface in $\mathbb{R}^{10}$. As well, for second countability I think that the fact that the surface is compact helps?
I would prefer hints. I am new to manifolds and want to learn the intuition for approaching problems by myself. What else should I notice about this space that may help me?