# 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:

$$\text{xymatrix}\mathrm{dom}(p)\text{ar}{[r]}^{f}\text{ar}{[d]}_{p}\mathrm{\&}\mathrm{dom}(q)\text{ar}{[d]}^{q}\{{p}_{0}\}\text{ar}{[r]}_{c}\mathrm{\&}\{{q}_{0}\}$$ |

## 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.

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.

Title | pointed set |
---|---|

Canonical name | PointedSet |

Date of creation | 2013-03-22 15:55:42 |

Last modified on | 2013-03-22 15:55:42 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 10 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 03E20 |

Synonym | base point |

Synonym | base-point |

Defines | basepoint |

Defines | pointed subset |