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The empty set is the unique initial object in Set, the category of sets. Every one-element set is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category.
The empty set serves as the initial object in Set with empty functions as morphisms. Every singleton is a terminal object, with the functions mapping all elements of the source sets to the single target element as morphisms. There are thus no zero objects in Set. The category Set is complete and co-complete.
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.
If C is a preadditive category, then every hom-set Hom(X,Y) is an abelian group and therefore has a zero element. These zero elements form a compatible family of zero morphisms for C making it into a category with zero morphisms. The category of sets does not have a zero object, but it does have an initial object, the empty set ∅.
The coproduct indexed by the empty set (that is, an empty coproduct) is the same as an initial object in . If J {\displaystyle J} is a set such that all coproducts for families indexed with J {\displaystyle J} exist, then it is possible to choose the products in a compatible fashion so that the coproduct turns into a functor C J → C ...
initial 1. An object A is initial if there is exactly one morphism from A to each object; e.g., empty set in Set. 2. An object A in an ∞-category C is initial if (,) is contractible for each object B in C. injective 1.
In object-oriented computer programming, a null object is an object with no referenced value or with defined neutral (null) behavior.The null object design pattern, which describes the uses of such objects and their behavior (or lack thereof), was first published as "Void Value" [1] and later in the Pattern Languages of Program Design book series as "Null Object".
The empty set (considered as a topological space) is the initial object of Top; any singleton topological space is a terminal object. There are thus no zero objects in Top. The product in Top is given by the product topology on the Cartesian product. The coproduct is given by the disjoint union of topological spaces.