It frustrates me when I see people talk of 'A null set' or 'An empty set'. One of the axioms of set theory is that two sets are equal if they contain the same elements. Since all null sets and empty sets contain the same elements [IE none at all] they are the same, so we refer to is as THE null set or THE empty set.

One way to think of the empty set is just through cardinality. The empty set has cardinality zero, no elements. Inset notation it's written { }. Nada, nothing. It's a subset of every single set.

In my philosophy, the empty set is the foundation of all mathematics. One is the cardinality of the set which contains the empty set, IE |{ {} }|=1

You just build up from nothing.