A basis for a space consists of elements of that space. So we are naturally led to how the property 'is a basis' interacts with operations ⊂ and ∩ and ∪. (Of course, a basis is actually a sequence in that it is ordered, but there is a natural extension of these operations.)
(a) Consider first how bases might be related by ⊂. Assume that U, W are subspaces of some vector space and that U ⊂ W. Can there exist bases BU for U and BW for W such that BU ⊂ BW? Must such bases exist?
For any basis BU for U, must there be a basis BW for W such that BU ⊂ BW?
For any basis BW for W, must there be a basis BU for U such that BU ⊂ BW?
For any bases BU, BW for U and W, must BU be a subset of BW?
(b) Is the ∩ of bases a basis? For what space?
(c) Is the ∪ of bases a basis? For what space?
(d) What about the complement operation? (Test any conjectures against some subspaces of R3.)