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

Extracting long basic sequences from systems of dispersed vectors