New sets from old

The main examples are the following:

Subsets:

 Given a set X and another set Y all elements of which also belong to X.
 Then Y is called a subset of X.
 The function iY: Y → X given by iY(x) = x is called the inclusion.

Products of sets:

 Given two sets X and Y.
 Then X×Y is the set of all pairs (a,b) of elements in X and Y respectively.
 The functions pX: X×Y → X and pY: X×Y → Y given by pX(a,b) = a, pY(a,b) = b are called the projections.

Quotients of sets:

 Given a set X and an equivalence relation ~ on X. 
 Then the quotient set X/~ is the set of equivalence classes {[x]: x∈X}.
 The function q: X → X/~ given by q(x) = [x] is called the quotient function.

Main applications:

• New topological spaces from old,
• New vector spaces from old,
• New complexes from old,
• New groups from old.