quotient topology

Open Map

A function

f : X \to Y

is said to be an open map if for every open set

U \in T_{X}

the set

f (U)

is open in

Y

The Projection Maps Are Open

The project maps:

π_{X} : X \times Y \to X

and

π_{Y} : X \times Y \to Y

are open

Let $M$ be an open set in $T_{X \times Y}$ therefore it is of the form $M = ⋃ A$ where $A \subseteq {U \times V : U \in T_{X}, V \in T_{Y}}$ .

But then $π_{X} (M) = π_{X} (⋃ A)$ = $⋃ {π_{X} (U_{A} \times V_{A}) : U_{A} \times V_{A} \in A} = ⋃ {U_{A} : U_{A} \times V_{A} \in A}$ so that $π_{X} (M)$ is a unions of open sets of $T_{X}$ and therefore is open in $X$ . Symetrically wealso have that $π_{Y}$ is open.

Quotient Map

Given an

(X, ~)

, then we define the quotient map of

~

as a function

π : X \to X ⧵ ~

(where we are mapping into the space modded out by the quotient)

π (x) = [x]

🏗️ ΘρϵηΠατπ🚧 (under construction)

🏗️ $Θ ρ ϵ η Π α τ π$ 🚧 (under construction)