In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. In axiomatic set theory it is postulated to exist by the axiom of empty set. From it all finite sets are constructed.
Notation
The standard notation for denoting the empty set, is the symbol or Ø, first used by the group of mainly French early 20thcentury mathematicians who wrote under the collective pseudonym of Nicolas Bourbaki. This should not be confused with the Greek letter Φ. Another common notation for the empty set is {}.
Properties
(Here we use mathematical symbols.)

For any set A, the empty set is a subset of A:
 ∀A: {} ⊆ A
 For any set A, the union of A with the empty set is A:
 ∀A: A ∪ {} = A
 For any set A, the intersection of A with the empty set is the empty set:
 ∀A: A ∩ {} = {}
 For any set A, the Cartesian product of A and the empty set is empty:
 ∀A: A × {} = {}
 The only subset of the empty set is the empty set itself:
 ∀A: A ⊆ {} ⇒ A = {}
 The number of elements of the empty set (that is its cardinality) is zero; in particular, the empty set is finite:
 {} = 0
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."
Common problems
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 bag containing its elements; an empty bag may be empty, but the bag itself certainly exists.
Some people balk at the first property listed above, 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."
Axiomatic set theory
In the axiomatization of set theory known as ZermeloFraenkel set theory, the existence of the empty set is assured by the axiom of empty set. The uniqueness of the empty set follows from the axiom of extensionality.
Any axiom that states the existence of any set will imply the axiom of empty set, using the axiom schema of separation. For example, if A is a set then the axiom schema of separation allows the construction of the set B = {x in A  x ≠ x}, which can be defined to be the empty set.
Does it exist or is it necessary?
While the empty set is a standard and universally accepted concept in mathematics, there are those who still entertain doubts.
Jonathan Lowe has argued that while the idea "was undoubtedly an important landmark in the history of mathematics, … we should not assume that its utility in calculation is dependent upon its actually denoting some object." It is not clear that such an idea makes sense. "All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and (3) is unique amongst sets in having no members. However, there are very many things that 'have no members', in the settheoretical sense—namely, all nonsets. It is perfectly clear why these things have no members, for they are not sets. What is unclear is how there can be, uniquely amongst sets, a set which has no members. We cannot conjure such an entity into existence by mere stipulation."
In "To be is to be the value of a variable…", Journal of Philosophy , 1984 (reprinted in his book Logic, Logic and Logic), the late George Boolos has argued that we can go a long way just by quantifying plurally over individuals, without reifying sets as singular entities having other entities as members.
In a recent book Tom McKay has disparaged the "singularist" assumption that natural expressions using plurals can be analysed using plural surrogates, such as signs for sets. He argues for an antisingularist theory which differs from set theory in that there is no analogue of the empty set, and there is just one relation, among, that is an analogue of both the membership and the subset relation.
Operations on the empty set
Operations performed on the empty set (as a set of things to be operated upon) can also be confusing. (Such operations are nullary operations.) For example, the sum of the elements of the empty set is zero, but the product of the elements of the empty set is one (see empty product). This may seem odd, since there are no elements of the empty set, so how could it matter whether they are added or multiplied (since “they” don't exist)? Ultimately, the results of these operations say more about the operation in question than about the empty set. For instance, notice that zero is the identity element for addition, and one is the identity element for multiplication.
The empty set and zero
It was mentioned earlier that the empty set has zero elements, or that its cardinality is zero. The connection between the two concepts goes further however: in the standard settheoretic definition of natural numbers, zero is defined as the empty set.
Category theory
If A is a set, then there exists precisely one function f from {} to A, the empty function. As a result, the empty set is the unique initial object of the category of sets and functions.
The empty set can be turned into a topological space in just one way (by defining the empty set to be open); this empty topological space is the unique initial object in the category of topological spaces with continuous maps.