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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:

  1. TVS's $V$ of larger and larger cardinality become more and more pathological / difficult / intractable or,

  2. 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.]

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    $\begingroup$ One result in this direction is the fact that a product of (non-trivial) bornological spaces is bornological if and only a product of lines over the same index set has this property. This is thus a property of the corresponding cardinality. $\endgroup$
    – priel
    Aug 22, 2015 at 12:57

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