# Null Set

In Axiomatic Set Theory, it is now more common to refer to the Empty Set, rather than to the Null Set, in order to avoid confusion with the concept of Null Set as used in Measure Theory.

The empty set is not the same thing as nothing; it is a set with nothing inside it, and a set is something. This often causes difficulty among those who first encounter it. It may be helpful to think of a set as a package containing its elements; an empty package may be empty, but the package itself certainly exists.

Some people balk at the first property of the empty set, namely, that the empty set is a subset of any set A. By the definition of subset, this claim means that for every element x of {}, x belongs to A. If it is not true that every element of {} is in A, there must be at least one element of {} that is not present in A. Since there are no elements of {} at all, there is no element of {} that is not in A, leading us to conclude that every element of {} is in A and that {} is a subset of A. Any statement that begins "for every element of {}" is not making any substantive claim; it is a vacuous truth. This is often paraphrased as "everything is true of the elements of the empty set."

The following are some mathematical truths about the empty set (denoted here as {}):

The empty set is NOT 0.

For any set A, the empty set is a subset of A;

For any set A, the union of A with the empty set is A;

For any set A, the intersection of A with the empty set is the empty set;

For any set A, the Cartesian product of A and the empty set is empty;

The only subset of the empty set is the empty set itself;

The number of elements of the empty set (that is its cardinality) is zero; in particular, the empty set is finite:

|{}| = 0

For any property:
• for every element of {} the property holds (vacuous truth)
• there is no element of {} for which the property holds

Conversely: if, for some property, the following two statements hold:
• for every element of V the property holds
• there is no element of V for which the property holds

then V = {}

Mathematicians speak of "the empty set" rather than "an empty set". In set theory, two sets are equal if they have the same elements; therefore there can be only one set with no elements.

Considered as a subset of the real number line (or more generally any topological space), the empty set is both closed and open. All its boundary points (of which there are none) are in the empty set, and the set is therefore closed; while all its interior points (of which there are again none) are in the empty set, and the set is therefore open. Moreover, the empty set is a compact set by the fact that every finite set is compact.

The closure of the empty set is empty. This is known as "preservation of nullary unions".