Underlying set

What do a group, a partially ordered set, and a topological space have in common? Each is a set with some structure built on top of it, and in each case, we call the set the underlying set. noteA group is a set with an operation, a poset is a set with an ordering, and a topological space is a set with a collection of subsets that satisfy a certain property.

Algebraic structures

An algebraic structure is a set equipped with operators that follow certain laws, such as a group, which is a pair \((X, \bullet)\) where \(X\) is a set and \(\bullet\) is an operator that follows certain laws. Given an algebraic structure, we can simply throw away the operators and recover the set ($X$, in this case), which is known as the “underlying set” of the structure.

The underlying set is sometimes known as the “carrier set.” Some algebraic structures have more than one underlying set; for example, a vector space is an algebraic structure built out of a field of scalars and a commutative group, in which case the term “underlying set” is ambiguous.