Let be a non-empty collection of bounded subsets of Then the -topology on is not altered if is replaced by any of the following collections of subsets of :
all subsets of all finite unions of sets in ;
all scalar multiples of all sets in ;
all finite Minkowski sums of sets in ;
the balanced hull of every set in ;
the closure of every set in ;
and if and are locally convex, then we may add to this list:
the closed convex balanced hull of every set in (en)
The -topology on is compatible with the vector space structure of if and only if every is -bounded;
that is, if and only if for every and every is bounded in (en)
If and if is a net in then in the -topology on if and only if for every converges uniformly to on (en)
