In Cox Little and Schenck, one result that is cited in chapter two is that if $D_n$ is the $n-dimensional$ torus, and $H < D_n$ is a closed subgroup then $H$ is itself a torus. Let the underlying field be $\mathbb{C}$.
The subgroup $H=\{(\pm 1, 1, 1....1) \}$ is a closed subgroup of $D_n$, being given by the ideal $\langle x_1^2-1, x_2-1,...x_n-1 \rangle$, but it is not even connected, let alone a torus.
Is my conclusion that Cox is wrong, right?