Proposition 4

In $\mathbf{R}^n$ ($n \geq 1$) the complement of every hyperplane has two connected components.

Let $g(x) = 0$ be an equation of a hyperplane $H$ in $\mathbf{R}^n$, $g$ being a linear polynomial. The set $CH$ is the union of the set $E_1$ of all points $x$ such that $g(x) > 0$ and the set $E_2$ of all points $x$ such that $g(x) < 0$. $E_1$ and $E_2$ are connected, for if $g(x) > 0$ and $g(y) > 0$ we have $g(ux + vy) = ug(x) + vg(y) > 0$ whenever $u \geq 0, v \geq 0$ and $u + v = 1$; in other words, the segment with end-points $x$ and $y$ is contained in $E_1$. Similarly for $E_2$. On the other hand, $CH$ is not connected, because its image in $\mathbf{R}$ under $g$ is the union of the intervals ]$0, +\infty$ [ and ]$-\infty, 0$ [.

The components $E_1, E_2$ of the complement of a hyperplane $H$ are called the open half-spaces determined by $H$.

The closures of $E_1$ and $E_2$, which are respectively $E_1 \cup H$ and $E_2 \cup H$, are called the closed half-spaces determined by $H$.

Observe that an affine mapping of $\mathbf{R}^n$ onto itself which transforms $H$ into a "coordinate" hyperplane, e.g., the hyperplane whose equation is $x_n = 0$, also transforms the open half-spaces determined by $H$ into the open half-spaces defined respectively by the relations $x_n > 0$ and $x_n < 0$; the latter are open boxes and therefore homeomorphic to $\mathbf{R}^n$.