Knowlpedia

Closure of intersections under an interior-point condition

If convex sets have intersecting interiors, closure distributes over their intersection

Theorem. Let XX be a normed vector space, and let Ω1,Ω2⊂X\Omega_1,\Omega_2\subset X be sets such that

int(Ω1)∩int(Ω2)≠∅.\mathrm{int}(\Omega_1)\cap \mathrm{int}(\Omega_2)\neq\emptyset.

Then

Ω1∩Ω2‾=Ω1‾∩Ω2‾.\overline{\Omega_1\cap\Omega_2}=\overline{\Omega_1}\cap\overline{\Omega_2}.
Remarks

Context. In general, A∩B‾⊂A‾∩B‾\overline{A\cap B}\subset \overline{A}\cap\overline{B} can be strict. Convexity plus an interior qualification condition forces equality, which is important in convex analysis and duality.

Proof idea. Use the existence of an interior point common to both sets to "stabilize" approximations and apply to build sequences in Ω1∩Ω2\Omega_1\cap\Omega_2 approximating any point in Ω1‾∩Ω2‾\overline{\Omega_1}\cap\overline{\Omega_2}.