lattice of subspaces
Collect ALL the subspaces of a space V into one structure and order them by inclusion: U <= W means U sits inside W. The smallest subspace is {0}, the largest is V itself, and in between sits a rich web of lines, planes, and hyperplanes nested in one another. This ordered collection is the lattice of subspaces — a single object that captures how V's pieces fit together.
A lattice is an ordered set where any two elements have a greatest lower bound (their meet) and a least upper bound (their join). For subspaces the meet of U and W is their intersection U ∩ W — the biggest subspace inside both. The join is their sum U + W — the smallest subspace containing both. Intersection and sum are thus not arbitrary operations but the lattice meet and join, dual to each other.
The deep fact is that this lattice is MODULAR: whenever U <= W, the modular law U + (X ∩ W) = (U + X) ∩ W holds for every X. This single identity is the engine behind the dimension formula dim(U + W) = dim U + dim W - dim(U ∩ W), and it explains why subspaces interlock so cleanly. The lattice is not distributive, though — that is exactly why two distinct lines and a third line can fail the naive set-style algebra.
Why care about the whole lattice rather than one subspace at a time? Because many questions are really about relationships: when is V a direct sum of pieces, which subspaces are invariant under an operator, how do the eigenspaces sit relative to each other. Thinking lattice-first replaces a pile of ad hoc inclusions with one organized structure you can reason about as a unit.
Intersection and sum are the lattice operations on subspaces; modularity links them to dimension.
The union of two subspaces is almost never a subspace — add a vector from one to a vector from the other and you usually leave the union. That failure is exactly why the lattice join is the SUM U + W, not the set union; the sum repairs the union by including all the combinations the union forgot.