pointed set

0.1 Definition

A pointed set is an ordered pair $(A,a)$ such that $A$ is a set and $a\in A$. The element $a$ is called the basepoint of $(A,a)$. At first glance, it seems appropriate enough to call any non-empty set a pointed set. However, the basepoint plays an important role in that if we select a different element $a^{\prime }\in A$, the ordered pair $(A,a^{\prime })$ forms a different pointed set from $(A,a)$. In fact, given any non-empty set $A$ with $n$ elements, $n$ pointed sets can be formed from $A$.

A function $f$ between two pointed sets $(A,a)$ and $(B,b)$ is just a function from $A$ to $B$ such that $f(a)=b$. Whereas there are ${|B|}^{\mid A\mid }$ functions from $A$ to $B$, only ${|B|}^{\mid A\mid -1}$ of them are from $(A,a)$ to $(B,b)$.

Pointed sets are mainly used as illustrative examples in the study of universal algebra as algebras with a single constant operator. This operator takes every element in the algebra to a unique constant, which is clearly the basepoint in our definition above. Any homomorphism (http://planetmath.org/HomomorphismBetweenAlgebraicSystems) between two algebras preserves basepoints (taking the basepoint of the domain algebra to the basepoint of the codomain algebra).

From the above discussion, we see that a pointed set can alternatively described as any constant function $p$ where the its domain is the underlying set, and its range consists of a single element $p_0\in \mathrm{dom}(p)$. A function $f$ from one pointed set $p$ to another pointed set $q$ can be seen as a function from the domain of $p$ to the domain of $q$ such that the following diagram commutes:

0.2 Creation of Pointed Sets from Existing Ones

Pointed Subsets. Given a pointed set $(A,a)$, a pointed subset of $(A,a)$ is an ordered pair $(A^{\prime },a)$, where $A^{\prime }$ is a subset of $A$. A pointed subset is clearly a pointed set.

Products of Pointed Sets. Given two pointed sets $(A,a)$ and $(B,b)$, their product is defined to be the ordered pair $(A\times B,(a,b))$. More generally, given a family of pointed sets $(A_i,a_i)$ indexed by $I$, we can form their Cartesian product to be the ordered pair $(\prod A_i,(a_i))$. Both the finite and the arbitrary cases produce pointed sets.

Quotients. Given a pointed set $(A,a)$ and an equivalence relation $R$ defined on $A$. For each $x\in A$, define $\overline{x}:=\{y\in A\mid yRx\}$. Then $A/R:=\{\overline{x}\mid x\in A\}$ is a subset of the power set $2^A$ of $A$, called the quotient of $A$ by $R$. Then $(A/R,\overline{a})$ is a pointed set.