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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 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:
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 $\overline{x}:=\lbrace y\in A \mid y R x\rbrace$ . Then $A/R:=\lbrace \overline{x}\mid x\in A\rbrace$ 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.
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![$\displaystyle \xymatrix{ {\operatorname{dom}(p)}\ar[r]^{f}\ar[d]_{p}&{\operatorname{dom}(q)}\ar[d]^{q}\ {\lbrace p_0\rbrace}\ar[r]_{c}&{\lbrace q_0\rbrace}} $](http://images.planetmath.org:8080/cache/objects/7936/js/img1.png "$\displaystyle \xymatrix{ {\operatorname{dom}(p)}\ar[r]^{f}\ar[d]_{p}&{\operatorname{dom}(q)}\ar[d]^{q}\ {\lbrace p_0\rbrace}\ar[r]_{c}&{\lbrace q_0\rbrace}} $")