Topological Quotients in Euclidean Space

# Topological Quotients in Euclidean Space

Recall from the Topological Quotients page that if $(X, \tau)$ is a topological space and $\sim$ is an equivalence relation on $X$ where for all $x \in X$, $[x] = \{ y \in X : x \sim y \}$ denotes the equivalence class of $x$ with respect to the relation $\sim$ and $X \: / \sim$ denotes the set of all equivalences classes on $X$, then the Quotient Topology is the final topology induced by the canonical quotient map $q : X \to X \: / \sim$.

We will now look at some examples of quotient topological spaces in Euclidean space with the respective usual topologies.

Consider the set $X = [0, 1]$.

Let $\tau_X$ is the usual topology consisting of open intervals from $X$ and consider the topological space $(X, \tau_x)$. Define an equivalence relation $\sim$ by $x \sim y$ if $x = y$ OR $x \sim y$ if $\{ x, y \} = \{0, 1 \}$. This equivalence relation can be thought of as "gluing" the endpoints of the closed interval $X$ together to form a circle:

Note that $\sim$ is indeed an equivalence relation. It is reflexive because $x \sim x$, symmetric since $x \sim y$ if and only if $y \sim x$, and transitive since $x \sim y$ and $y \sim z$ implies that $x \sim z$. Now consider the set of equivalence classes $X \: / \sim$:

(1)
\begin{align} \quad X \: / \sim \: = \{ \{ x \} : x \in [0, 1] \} \cup \{ \{0, 1\} \} \end{align}

Consider the canonical quotient map $q : X \to X \: / \sim$ defined for all $x \in [0, 1]$ by $q(x) = [x]$. Then the final topology induced by $q$ is given by:

(2)
\begin{align} \quad \tau_{X / \sim} = \{ q^{-1}(U) : U \in \tau_X \} \end{align}

So the topology on $\tau_{X / \sim}$ can be described somewhat vaguely as unions of "open arcs" not containing $0$ or $1$ as illustrated below:

For another example of a quotient topological space, let $X = [0, 1] \times [0, 1]$.

Consider the topological space $([0, 1] \times [0, 1], \tau_X)$ where $\tau_X$ is the usual topology whose open sets are generated by open disks in $\mathbb{R}^2$.

Define an equivalence relation $\sim$ on $X$ by $(x, y) \sim (w, z)$ if $(x, y) = (w, z)$, $y = z$ and $\{x, w \} = \{0 , 1 \}$, or $x = w$ and $\{ y, z \} = \{0 , 1 \}$.

By "gluing" equivalent points together, we can visualize $X \: / \sim$ with the following diagram:

The topology on $X \: / \sim$ will be generated by "open subsurfaces" of the torus above that do not intersect the "glue-lines".