  Let $V$ be a real or complex vector space. The finite topology of $V$
consists of all the subsets $U$ for which the intersection $U \cap F$ is closed
in $F$ for every finite-dimensional linear subspace of $V$. It is known that if
$V$ has countable dimension, then this topology coincides with the greatest
vector space topology and with the greatest locally convex vector space
topology. Here, we note that in this case the finite topology is simply the
union of all the normed space topologies.
  In connection to this problem, we characterize the vector spaces on which
every family of norms with given cardinality is dominated.
