Finest chains of subspaces in infinite-dimensional vector spaces
Suppose that \(K\) is a set of subspaces of a vector space \(V\), such that \(\{0\}\), \(V\in K\) and such that \(K\) is totally ordered by inclusion. Such a \(K\) is called a chain of subspaces. A finest chain \(K\) is maximal, i.e. no subspaces can be added to \(K\) without destroying the total order. Clearly, if \(V\) is \(d\)-dimensional \((d<\aleph_0)\), then each finest chain has cardinality \(d+1\). It is shown that this result remains true for \(d=\aleph_0\), but fails to hold in case of uncountable dimension. For example, if \(F\) is a field, then the vector space of all mappings \(\mathbb{N}\to F\) has no countable basis, but a countable finest chain of subspaces.
scientific article; zbMATH DE number 1036098