# 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 $p_{0}\in\operatorname{dom}(p)$. 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:

 $\xymatrix{{\operatorname{dom}(p)}\ar[r]^{f}\ar[d]_{p}&{\operatorname{dom}(q)}% \ar[d]^{q}\\ {\{p_{0}\}}\ar[r]_{c}&{\{q_{0}\}}}$

## 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.

. 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.

Title pointed set PointedSet 2013-03-22 15:55:42 2013-03-22 15:55:42 CWoo (3771) CWoo (3771) 10 CWoo (3771) Definition msc 03E20 base point base-point basepoint pointed subset