empty family

Given any set $X$, there is a unique **empty family** of elements of $X$. Formally, this is given by the empty function to $X$, the unique function to $X$ from the empty set. As the empty set is finite and (a fortiori) countable, this empty family counts as a list (or tuple) and a stream; in such a guise it is known as the **empty list** (or **$0$-tuple**) or the **empty stream** (or *empty sequence* if the term ‘sequence’ is used in a sufficiently general sense).

When treating it as an element of the free monoid on $X$, the empty list may be written $()$, $*$, or $\epsilon$, perhaps with a subscript $X$ if desired. (Thinking of all empty lists, regardless of the underlying set, as equal is like thinking of all empty subsets as equal.)

Besides empty families of elements of sets, we have the notions of the empty family of elements of a given proper class, the empty family of elements of a given preset, the empty family of term of a given type, the empty family of objects and the empty family of morphisms of a given category, and more generally the empty family of whatever you want.

Last revised on April 6, 2017 at 00:07:48. See the history of this page for a list of all contributions to it.