There is a familiar notion and construct written “∪x” where x is a set.
B belongs to this just if B belongs to something that belongs to x.
b ∈ ∪x ↔ ∃z (b ∈ z ∧ z ∈ x).
It would seem natural to define a similar notion to go with ∩x. Perhaps B belongs to this just if it belongs to every thing that belongs to x:
b ∈ ∩x ↔ ∀z (z ∈ x → b ∈ z).
This leads however to the awkward result that ∀b(b ∈ ∩∅). One might ask for a dispensation on the meaning of ∩∅ but that gets you into other problems. Note the convenient theorem:
A⊂B → ∪A ⊂ ∪B.
Its analog would be
A⊂B → ∩B ⊂ ∩A.
∩∅ is too much like 1/0. Set theories that admit the universal set can ascribe that to ∩∅. Such theories are known to be inconsistent or otherwise very inconvenient.
∩ thus remains a troubled concept and notation. The idea of the intersection of two set, x∩y, is not problematical. The same trouble arises in intersections of sets of subspaces of vector spaces. If the given set of spaces is empty, the answer is the ‘whole space’ but what indeed is that? Must it be guessed from context? The trouble with ∩ is that it is contravariant.