We show that for a non-flat bornological space there is always a bornological countable enlargement; moreover, when the space is non-flat and ultrabornological the countable enlargement may be chosen to be both bornological and barrelled. It is also shown that countable enlargements for barrelled or bornological spaces are always Mackey topologies, and every quasibarrelled space that is not barrelled has a quasibarrelled countable enlargement.
AMS 2000 Mathematics subject classification: Primary 46A08; 46A20
