structural:summarization, stylistic:style_transfer
Category theory

If A {\displaystyle A} is a set, then there exists precisely one function f {\displaystyle f} from ∅ {\displaystyle \varnothing } to A, {\displaystyle 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.