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 from to $X$ from the empty set. As the empty set is finite and (a fortiori) countable, this empty family counts as a list and a sequence; in such a guise it is known as the **empty list** or the **empty sequence**.

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.

Similarly, we have the notions of the empty family of elements of a preset or other notion of 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.

Revised on August 19, 2010 18:05:38
by Toby Bartels
(64.89.61.32)