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:<math>\mathcal{G}</math> is assumed to be closed with respect to the formation of subsets of finite unions of sets in <math>\mathcal{G}</math> (i.e. every subset of every finite union of sets in <math>\mathcal{G}</math> belongs to <math>\mathcal{G}</math>).
Some authors (e.g. Trèves {{sfn|Trèves|2006|Section 32}}) require that <math>\mathcal{G}</math> be directed under subset inclusion and that it satisfy the following condition:
:If <math>G \in \mathcal{G}</math> and <math>s</math> is a scalar then there exists a <math>H \in \mathcal{G}</math> such that <math>s G \subseteq H.</math>
If <math>\mathcal{G}</math> is a [[bornology]] on <math>X,</math> which is often the case, then these axioms are satisfied.
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