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 for every one of its points (of which there are again none), there is an open neighbourhood 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."
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.
By the definition of subset, the empty set is a subset of any set A, as every element x of {} 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."- Wikipedia.
