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The above assumption guarantees that the collection of sets <math>\mathcal{U}(G, N)</math> forms a [[filter base]].
The next assumption will guarantee that the sets <math>\mathcal{U}(G, N)</math> are [[Balanced set|balanced]].
Every TVS has a neighborhood basis at 0 consisting of balanced sets so this assumption isn't
:'''Assumption''' (<math>N \in \mathcal{N}</math> are balanced): <math>\mathcal{N}</math> is a neighborhoods basis of the origin in <math>Y</math> that consists entirely of [[Balanced set|balanced]] sets.
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