Interior and closure of a convex set are convex

Proposition. Let $X$ be a normed vector space and let $\Omega\subset X$ be convex . Then:

• the interior $\mathrm{int}(\Omega)$ is convex (possibly empty),
• the closure $\overline{\Omega}$ is convex.

Context. Convexity is stable under two basic topological operations in normed spaces, which is essential for analyzing convex feasible regions.

Proof sketch. For $\overline{\Omega}$, approximate points by sequences in $\Omega$ and use convexity plus limit arguments. For $\mathrm{int}(\Omega)$, if $x,y\in\mathrm{int}(\Omega)$ then small balls around $x$ and $y$ lie in $\Omega$; convex combinations of these balls give a neighborhood of $\lambda x+(1-\lambda)y$ contained in $\Omega$, so the combination lies in $\mathrm{int}(\Omega)$.