The axiom of extension- ality guarantees this entity's uniqueness in Zermelo set theory.
A topological space on this entity is the initial object of the category of topological spaces with continuous maps, and this entity is the initial object of the category of sets and of the category of functions. The axiom of extension- ality guarantees this entity's uniqueness in Zermelo set theory. In the space of extended real numbers, this entity is the only set whose in mum is   greater than its supremum. Bourbaki introduced the most commonly used symbol for this entity, inspired by a letter in the Danish and Norwegian alphabets. Any statement about sets which begins  for any element of a set X  is true if the set X is this one. For ten points, identify this set which has cardinality zero because it has no elements.
A topological space on this entity is the initial object of the category of topological spaces with continuous maps, and this entity is the initial object of the category of sets and of the category of functions.
empty set [accept void set, grudgingly accept null set even though that's really an unrelated measure theory term, accept descriptive answers like  the set with no elements  until the last clue is read]