A pointed set is an ordered pair such that is a set and . The element is called the basepoint of . 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 , the ordered pair forms a different pointed set from . In fact, given any non-empty set with elements, pointed sets can be formed from .
A function between two pointed sets and is just a function from to such that . Whereas there are functions from to , only of them are from to .
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 where the its domain is the underlying set, and its range consists of a single element . A function from one pointed set to another pointed set can be seen as a function from the domain of to the domain of such that the following diagram commutes:
0.2 Creation of Pointed Sets from Existing Ones
Pointed Subsets. Given a pointed set , a pointed subset of is an ordered pair , where is a subset of . A pointed subset is clearly a pointed set.
Products of Pointed Sets. Given two pointed sets and , their product is defined to be the ordered pair . More generally, given a family of pointed sets indexed by , we can form their Cartesian product to be the ordered pair . Both the finite and the arbitrary cases produce pointed sets.
|Date of creation||2013-03-22 15:55:42|
|Last modified on||2013-03-22 15:55:42|
|Last modified by||CWoo (3771)|