pointed set

0.1 Definition

A pointed set is an ordered pair (A,a) such that A is a set and aA. 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 aA, the ordered pair (A,a) 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|A functions from A to B, only |B|A-1 of them are from (A,a) to (B,b).

Pointed sets are mainly used as illustrative examples in the study of universal algebraMathworldPlanetmathPlanetmath as algebrasMathworldPlanetmathPlanetmathPlanetmath 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 homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (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 p0dom(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,a), where A is a subset of A. A pointed subset is clearly a pointed set.

ProductsMathworldPlanetmathPlanetmathPlanetmath of Pointed Sets. Given two pointed sets (A,a) and (B,b), their product is defined to be the ordered pair (A×B,(a,b)). More generally, given a family of pointed sets (Ai,ai) indexed by I, we can form their Cartesian product to be the ordered pair (Ai,(ai)). Both the finite and the arbitrary cases produce pointed sets.

QuotientsPlanetmathPlanetmath. Given a pointed set (A,a) and an equivalence relationMathworldPlanetmath R defined on A. For each xA, define x¯:={yAyRx}. Then A/R:={x¯xA} is a subset of the power setMathworldPlanetmath 2A of A, called the quotient of A by R. Then (A/R,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