  Spectrahedra are sets defined by linear matrix inequalities. Projections of
spectrahedra are called semidefinitely representable sets. Both kinds of sets
are of practical use in polynomial optimization, since they occur as feasible
sets in semidefinite programming. There are several recent results on the
question which sets are semidefinite representable. So far, all results focus
on the case of closed sets. In this work we develop a new method to prove
semidefinite representability of sets which are not closed. For example, the
interior of a semidefinite representable set is shown to be semidefinite
representable. More general, one can remove faces of a semidefinite
representable set and preserve semidefinite representability, as long as the
faces are parametrized in a suitable way.
