Filter on Set is Proper Filter/Note about Axioms

Filter on Set is Proper Filter Note about Axioms

It seems at first glance that the condition $T \in \FF$ is not axiomatic, as it is clear from the third property:

$U \in \FF: U \subseteq T \subseteq T \implies T \in \FF$

However, one of the properties of a filter is that it is specifically not empty.

Specifying that $T \in \FF$ is therefore equivalent to specifying that $\FF \ne \O$.

Thus it would be possible to cite the first axiom as $\FF \ne \O$ instead, but this is rarely done.