Topology: An Inquiry-Based Approach

We are familiar with the word quotient when working with rational numbers. That is, the fraction $\frac{1}{2}$ is the quotient of $1$ by $2\text{,}$ and the set of rational numbers is the collection of all defined quotients of integers. The word quotient seems to have come from the latin word “quotiens”, which can be translated as “how often” or “how many times”. We can think of the fraction $\frac{1}{2}$ as dividing a unit ($1)$ into two pieces. So we often apply the word quotient to any kind of construction that divides a set into pieces. Another familiar quotient construction is the set ${\mathbb{Z}}_n\text{,}$ the set of quotients of integers after we divide by $n\text{.}$ Another way to think of ${\mathbb{Z}}_n$ is as a quotient $\mathbb{Z}/n\mathbb{Z}\text{,}$ where $n\mathbb{Z}$ is the set of multiples of $n$ and two integers $a$ and $b$ are identified with each other (or are equivalent) if $b-a\in n\mathbb{Z}\text{.}$ This defines the relation of congruence module $n$ on $\mathbb{Z}\text{,}$ and the elements of $\mathbb{Z}/n\mathbb{Z}$ are the equivalence classes for this relation. We make similar constructions in many branches of mathematics by defining an equivalence relation on a set, and we then divide the set into pieces (the equivalence classes) and call the set of equivalence classes a quotient space. We explore the concept of quotient spaces of topological spaces in this section.

As an example, take the interval $X=[0,1]$ in $\mathbb{R}$ and bend it to be able to glue the endpoints together. The resulting object is a circle. By identifying the endpoints $0$ and $1$ of the interval, we are able to create a new topological space. We can view this gluing or identifying of points in the space $X$ in a formal way that allows us to recognize the resulting space as a quotient space.