Math4201 Topology I (Lecture 17)

Quotient topology

How can we define topologies on the space obtained points in a topological space?

Quotient map

Let $(X,\mathcal{T})$ be a topological space. $X^*$ is a set and $q:X\to X^*$ is a surjective map.

The quotient topology on $X^*$ is defined as follows:

$U\subseteq X^*$ is open if and only if $q^{-1}(U)$ is open in $X$.

In particular, $q$ is continuous map.

Definition of quotient map

$q:X\to X^*$ defined above is called a quotient map.

Definition of quotient space

$(X^*,\mathcal{T}^*)$ is called the quotient space of $X$ by $q$.

Typical way of constructing a surjective map

Equivalence relation

$\sim$ is a subset of $X\times X$ satisfying:

• reflexive: $\forall x\in X, x\sim x$
• symmetric: $\forall x,y\in X, x\sim y\implies y\sim x$
• transitive: $\forall x,y,z\in X, x\sim y\text{ and } y\sim z\implies x\sim z$

Equivalence classes

Check equivalence relation.

For $x\in X$, the equivalence class of $x$ is denoted as $[x]\coloneqq \{y\in X\mid y\sim x\}$.

$X^*$ is the set of all equivalence classes on $X$.

$q:X\to X^*$ is defined as $q(x)=[x]$ will be a surjective map.

We will show that the quotient topology is a topology on $X^*$.