Splitting $\mathbb{R}^n$ into two subspaces

Question

Does a plane in a more than $3$-dimensional space split it into two connected components?

If not, what is the dimension of subspace of $\mathbb R^n$ which splits it into two connected components? Is it $\mathbb R^{n-1}$?

Answer 1

Yes, your guess is right, it's $\mathbb R^{n-1}$.

Given a $n-1$-dimensional subspace $V$ of $\mathbb R^n$, there exists a linear functional $f\colon\mathbb R^n\to\mathbb R$ such that $\ker f = V$. In that way, $V$ splits $\mathbb R^n$ in two connected components, namely $V_+ = \{x\in\mathbb R^n : f(x) > 0 \}$ and $V_- = \{x\in\mathbb R^n : f(x) < 0 \}$.