  We study Banach spaces satisfying some geometric or structural properties
involving tightness of transfinite sequences of nested linear subspaces. These
properties are much weaker than WCG and closely related to Corson's property
(C). Given a transfinite sequence of normalized vectors, which is dispersed or
null in some sense, we extract a subsequence which is a biorthogonal sequence,
or even a weakly null monotone basic sequence, depending on the setting. The
Separable Complementation Property is established for spaces with an M-basis
under rather weak geometric properties. We also consider an analogy of the
Baire Category Theorem for the lattice of closed linear subspaces.
