Separation by a Hyperplane

Let $X$ be a real vector space and let $\Omega_1,\Omega_2\subset X$ be nonempty.

We say that $\Omega_1$ and $\Omega_2$ can be separated by a hyperplane if there exists a nonzero linear functional $f:X\to\mathbb{R}$ such that

Geometrically, picking any $\alpha$ with $\sup_{\Omega_1}f\le \alpha\le \inf_{\Omega_2}f$ produces the hyperplane $\{x\mid f(x)=\alpha\}$ lying between the two sets; see the sup/inf characterization .