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:


\mathcal{T}^* = \{U\subseteq X^*\mid q^{-1}(U)\in \mathcal{T}\}

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.

Example of surjective maps and their quotient spaces

Let X=\mathbb{R}^2 and (s,t)\sim (s',t') if and only if s-s' and t-t' are both integers.

This space as a topological space is homeomorphic to the torus.

Let X=\{(s,t)\in \mathbb{R}^2\mid s^2+t^2\leq 1\} and (s,t)\sim (s',t') if and only if s^2+t^2 and s'^2+t'^2. with subspace topology as a subspace of \mathbb{R}^2.

This space as a topological space is homeomorphic to the spherical shell S^2.

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

Proof

We need to show that the quotient topology is a topology on X^*.

\emptyset, X^* are open in X^*.

\emptyset, X^* are open in X^* because q^{-1}(\emptyset)=q^{-1}(X^*)=\emptyset and q^{-1}(X^*)=X are open in X.

\mathcal{T}^* is closed with respect to arbitrary unions.


q^{-1}(\bigcup_{\alpha \in I} U_\alpha)=\bigcup_{\alpha \in I} q^{-1}(U_\alpha)

\mathcal{T}^* is closed with respect to finite intersections.


q^{-1}(\bigcap_{\alpha \in I} U_\alpha)=\bigcap_{\alpha \in I} q^{-1}(U_\alpha)

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