In this paper a numeric criterion, k( m, p) ⩾ n, is given for stabilizability by constant gain output feedback of the generic linear multivariable system with m inputs, n states, and p outputs. This criterion is defined in terms of a topological invariant of the space of gains, both finite and infinite, arising from an interpretation of the problem of stabilizability as a problem concerning open covers of this space. This invariant, originally considered by Ljusternik and Šnirel'mann in the calculus of variations, can be estimated from below by using a theorem of Eilenberg together with the methods of the Schubert calculus, thus leading to some very explicit corollaries concerning generic stabilizability. As far as I am aware, these corollaries cannot be derived from existing results concerning pole-assignability. After passing to a problem on coverings of the (compactified) gain space, the main technical problem which remains in a high gain stability analysis — stated here as a High Gain Lemma — which appears to be of independent interest. That is, the topological argument implies that the closed-loop system is ‘stable’ for some gain, possibly infinite. If the root-locus map were defined and continuous at all infinite gains, then the conclusion that the system is stabilizable by finite gain could be deduced from a simple continuity argument. However, it is known that if mp > n then the root-locus map has points of discontinuity at infinity, and, since mp ⩾ n is now known to be necessary for generic stabilizability, in most cases of interest one requires a far more subtle argument than one would give, for example, in the scalar case.