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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 closedin $F$ for every finite-dimensional linear subspace of $V$. It is known that if$V$ has countable dimension, then this topology coincides with the greatestvector space topology and with the greatest locally convex vector spacetopology. Here, we note that in this case the finite topology is simply theunion of all the normed space topologies.  In connection to this problem, we characterize the vector spaces on whichevery family of norms with given cardinality is dominated.

On the finite topology of a vector space and the domination problem for families of norms