In any topological space "X", the empty set is open by definition, as is "X". Since the complement of an open set is closed and the empty set and "X" are complements of each other, the empty set is also closed, making it a clopen set. Moreover, the empty set is compact 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." 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, called the empty 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. In fact, it is a strict initial object: only the empty set has a function to the empty set. In the von Neumann construction of the ordinals, 0 is defined as the empty set, and the successor of an ordinal is defined as formula_24. Thus, we have formula_25, formula_26, formula_27, and so on. The von Neumann construction, along with the axiom of infinity, which guarantees the existence of at least one infinite set, can be used to construct the set of natural numbers, formula_28, such that the Peano axioms of arithmetic are satisfied. In Zermelo set theory, the existence of the empty set is assured by the axiom of empty set, and its uniqueness follows from the axiom of extensionality. However, the axiom of empty set can be shown redundant in at least two ways:  While the empty set is a standard and widely accepted mathematical concept, it remains an ontological curiosity, whose meaning and usefulness are debated by philosophers and logicians. The empty set is not the same thing as "nothing"; rather, it is a set with nothing "inside" it and a set is always "something". This issue can be overcome by viewing a set as a bag—an empty bag undoubtedly still exists. Darling (2004) explains that the empty set is not nothing, but rather "the set of all triangles with four sides, the set of all numbers that are bigger than nine but smaller than eight, and the set of all opening moves in chess that involve a king."