Start with the empty set. We define it as “the set of all objects not equal to themselves,” since there are no such objects. All empty sets have the same members—no members at all! Therefore, as sets they're identical, by definition of identity of sets. In other words, there's only one empty set. This unique empty set is our building block. Next comes the set whose only member is—the empty set. This set is not empty.

The existence of the empty set is entailed, in the presence of the axiom scheme of separation, by the existence of any set whatever.

A topological space may be very sparsely endowed with open sets. As we know, some spaces have only two, the empty set and the full space.

Their existence can be generated on the basis of the empty set, since one is simply the set of the empty set, two is simply that set plus the empty set and so forth.

Empty set

Meaning

Noun

Synonyms