(Preservation of closeness result). Let be a nonempty subset of and let be an matrix.

1. If then the set is closed.

2. Let be a nonempty subset of given by linear constraints If then the set is closed.

3. Let is given by the quadratic constraints where the are positive semidefinite matrices. Then the set is closed.

(1). Let , where the is the ball around of radius . The sets are nested if . It is suffice to prove that is not empty for any sequence

We have Therefore, by the proposition ( Recession cone of inverse image ), Consequently, in the context of the proposition ( Principal intersection result ) for , Since, generally to accomplish the condition of the ( Principal intersection result ) it is enough to have as required by the theorem.

(2). Let . We introduce the sets for and aim to prove that the intersection is not empty.

We have By the propositions ( Recession cone of inverse image ) and ( Recession cone of intersection ) Consequently, in the context of the proposition ( Principal intersection result ) for , Since, generally to accomplish the condition of the ( Principal intersection result ) it is enough to have

(3). Let . We introduce the sets for and aim to prove that the intersection is not empty.

We have

We now apply the proposition ( Quadratic intersection result ) to conclude the proof.