Quotient Topology

notations

top: topological space (structure)
hom: homomorphism on top

1. Definition

Quotient Space 1

Suppose X is a top, f:X→Y. Quotient topology of Y is {B⊂Y|f−1[B]∈X} denoted as X/f. Under this topology Y is the quotient space of X.

Quotient topology is the finest top for Y that f is continous, meanwhile f is a quotient map. Remember f−1[A]∈X⟺A∈X/f

Quotient Space 2

Suppose X is a top, ∼ is an equivalent relation (or partition), let f(x)=[x] (equivalent class or component). Quotient space of X under ∼ is X/f={A|⋃x∈A[x]} denoted as X/∼.

Remark

A partition π gives an equivalent relation x∼y iff x,y∈A∈π. Define X/π=X/∼

Remark

if f is open, then f is a quotient map, not the other way. In topological group, the quotient map is indeed open.

Gluing space

A⊂X, given a partition π={A,{x},{y},⋯}, define gluing space X/A=X/π. Element {x} in X/A is denoted as x for convenience.

To glue two spaces

X∩Y=∅, x∈X,y∈Y, then X+x,yY=X∪Y/{x,y}.

2. Basic Theorems

Theorem 1

If f:X→Y is a quotient map, then ϕ(y)=f−1[y]:Y≃X/f.