Subspaces of vector spaces are kind of like closed sets. They are closed under intersection. There is the concept of span (for any collection of vectors) that gives you a subspace and closure (for any collection of points) that gives you a closed set. For any other vector in the space, we can scale it up and not touch the subspace; for any other point in the space, we can take a neighborhood of it and not touch the closed set.
My question is: is this a consequence somehow of definitions? That if we were to express all definitions in formal logic, the parallels would be completely obvious?
Apologies if this is the wrong forum to be asking such a question.
Both the closed subsets of a topological space and linear subspaces of a vector space form a lattice (actually, a complete lattice). Furthemore, both cases can be seen as special cases of lattices of closed elements of some closure operator: in the topological case the operator is the actual closure, and in the linear case the closure operator is the span.