pointed set

pointed set

0.1

Definition

A

pointed set

is an

ordered pair

(

A

,

a

)

such that

A

is a set and

a

∈

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

′

∈

A

, 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 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

∈

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

⁢

dom

⁡

(

p

)

⁢

\ar

⁢

[

r

]

f

⁢

\ar

⁢

[

d

]

p

⁢

&

⁢

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

′

,

a

)

, where

A

′

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

×

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

(

∏

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

∈

A

, define

x

¯

:=

{

y

∈

A

∣

y

⁢

R

⁢

x

}

. Then

A

/

R

:=

{

x

¯

∣

x

∈

A

}

is a subset of the

power set

2

A

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

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