Given a topological vector space $V$, let its density be the smallest cardinal $A$ such that a set of cardinality $A$ is dense in $V$. Naively, it seems one of two things happen:
TVS's $V$ of larger and larger cardinality become more and more pathological / difficult / intractable or,
There is a cardinal $A_{max}$ so that almost all pathological behavior exhibited in any TVS is exhibited in a TVS of density $A$ less than or equal to $A_{max}$.
More concretely, are there other cardinality related restrictions one can put on a TVS to obtain decent results, or are such pursuits apparently fruitless? Is any cardinality restriction looser than separability useful?
[Note: Model Theory would have a lot to say about this, but I'm looking for additional perspectives as well. In addition, we could ask the same question about other structures, but I'd like to start with something concrete.]