A geometric framework for robust stabilization of infinite-dimensional time-varying linear systems is presented. The uncertainty of a system is described by perturbations of its graph and is measured in the gap metric. Necessary and sufficient conditions for robust stability are generalized from the time-invariant case. An example is given to highlight an important difference between the obstructions, which limit the size of a stabilizable gap ball, in the time-varying and time-invariant cases. Several results on the gap metric and the gap topology are established that are central in a geometric treatment of the robust stabilizability problem in the gap. In particular, the concept of a `graphable' subspace is introduced in the paper. Subspaces that fail to be graphable are characterized by an index condition on a certain semi-Fredholm operator.