Identification Spaces in Quotient Topology

Introduction: Why identify points at all? 🌍

Imagine a paper map, a sheet of cardboard, or a clay model in your hands, students. One of the most powerful ideas in topology is that we can change a space by gluing points together. This is called forming an identification space. Instead of studying every point separately, we decide that some points should count as the same point. That simple idea creates many important spaces used in mathematics, science, and engineering.

In this lesson, you will learn how identification spaces work, why they matter, and how they connect to quotient topology. By the end, you should be able to explain the basic terms, describe how the topology is built, and work through classic examples like making a circle from an interval or creating a torus from a square.

Learning objectives

Explain the main ideas and terminology behind identification spaces.
Apply topological reasoning to examples of identification spaces.
Connect identification spaces to quotient topology.
Summarize how identification spaces fit into the broader study of topology.
Use examples and evidence to describe how points are identified in a space.

What is an identification space? 🧩

An identification space begins with a topological space $X$ and an equivalence relation $\sim$ on $X$. The relation says which points are considered equivalent, or in other words, which points are to be identified.

If $x \sim y$, then the points $x$ and $y$ are treated as the same point in the new space. The set of all points equivalent to $x$ is called the equivalence class of $x$, written as $[x]$.

The new space is the set of all equivalence classes, often written as $X/\sim$. This is the set we get after the identification is made.

A key idea is that the new space is not just a set. It also gets a topology, called the quotient topology. This topology tells us which sets are open in the identification space.

The quotient map

The natural function from $X$ to the identification space $X/\sim$ is called the quotient map and is usually written as $q:X\to X/\sim$. It sends each point $x\in X$ to its equivalence class $[x]$.

This map is always surjective, because every equivalence class comes from at least one point in $X$.

The quotient topology on $X/\sim$ is defined so that a set $U\subseteq X/\sim$ is open if and only if $q^{-1}(U)$ is open in $X$. This rule is what makes quotient spaces so important: it transfers the topology from the original space to the new one in a precise way.

How the topology is built 🔧

The quotient topology is designed to make the quotient map behave well. Since the map $q$ is the bridge between the original space and the identification space, we define openness using $q^{-1}$.

A set $U$ in the quotient space is open exactly when its preimage under $q$ is open in the original space. This may feel backward at first, but it is very useful. It lets us understand the new space by checking ordinary open sets in the original space.

This rule also means that the quotient map $q$ is a continuous map by construction. In fact, it is the standard example of a map whose target space is defined using continuity.

Example 1: Making a circle from an interval

Take the closed interval $[0,1]$. Now identify the endpoints by saying $0\sim 1$, while every other point stays separate. The result is a space that is topologically the same as a circle $S^1$.

Why does this make sense? If you walk from $0$ to $1$ on the interval and then glue the ends together, the two ends meet, forming a loop. Topologically, a circle has no boundary points, and the identification removes the boundary of the interval.

So the quotient space $[0,1]/\sim$ where $0\sim 1$ is a standard example of an identification space.

Example 2: Turning a square into a torus

A torus can be formed by taking a square and identifying opposite edges. More specifically, points on the left edge are matched with corresponding points on the right edge, and points on the top edge are matched with corresponding points on the bottom edge.

This is a famous identification space. Although the square itself is flat, the identifications create the surface of a doughnut-like shape. This example shows how quotient topology can create spaces that are hard to visualize directly but are easy to define using point identifications.

Why equivalence relations matter

The equivalence relation $\sim$ is the rule that decides which points are glued together. It must satisfy three conditions:

Reflexive: $x\sim x$ for every $x\in X$
Symmetric: if $x\sim y$, then $y\sim x$
Transitive: if $x\sim y$ and $y\sim z$, then $x\sim z$

These conditions guarantee that the points split into well-defined equivalence classes. Without them, the identification would not be consistent.

Example 3: Collapsing a subspace to a point

Suppose $X$ is a space and $A\subseteq X$ is a subspace. We can identify every point of $A$ with each other, while leaving points outside $A$ unchanged. The result is a space where the entire set $A$ becomes a single point.

This idea appears often in topology. For example, if you collapse the boundary of a disk to one point, you get a space related to the sphere. Identification spaces help convert complex shapes into simpler ones by shrinking parts of them.

What do open sets look like? 🔍

In an identification space, open sets are not chosen by looking directly at the classes alone. Instead, we check their preimages in the original space.

Suppose $U\subseteq X/\sim$. To decide whether $U$ is open, we look at $q^{-1}(U)$ in $X$. If $q^{-1}(U)$ is open in $X$, then $U$ is open in the quotient space.

This matters because a set in the quotient space may look strange if viewed directly. But its preimage may be a familiar open set in $X$.

A useful way to think about it

You can think of the quotient space as a new world created from the old one. The quotient map says which old points became one new point. Open sets in the new world are exactly the ones whose “shadow” in the old world was open.

This is one reason quotient topology is such a natural construction. It preserves the idea of continuity while allowing spaces to be built by gluing.

Connection to quotient topology

Identification spaces are not separate from quotient topology; they are one of its central uses.

A quotient topology is any topology obtained from a surjective map $q:X\to Y$ such that a set $U\subseteq Y$ is open if and only if $q^{-1}(U)$ is open in $X$. When the points of $X$ are identified by an equivalence relation, the resulting space $X/\sim$ with the quotient topology is called an identification space.

So the relationship is:

start with a space $X$
choose an equivalence relation $\sim$
form the set of classes $X/\sim$
give it the quotient topology using the map $q:X\to X/\sim$

This is one of the first major constructions in topology because it lets us build new spaces from old ones in a controlled way.

Why this construction is powerful

Many spaces in topology are easier to describe by gluing than by writing down all their open sets from scratch. Identification spaces give a clean language for this process.

For example, surfaces can often be described by polygons with edges identified in specific ways. This is how mathematicians classify and study objects like the torus, the projective plane, and the Klein bottle.

Common mistakes to avoid ⚠️

A frequent mistake is thinking that if two points are identified, then the quotient space is just the original space with those points “removed.” That is not correct. The points are not removed; they are merged into one point.

Another mistake is assuming that the quotient topology is the same as the original topology. It is usually different, because the new space has a new set of points and a new notion of openness.

It is also important not to confuse the equivalence classes with open sets. A class $[x]$ is the set of points identified with $x$, but it is not necessarily open.

Conclusion

Identification spaces are a core idea in quotient topology, students. They let us build new spaces by declaring certain points to be the same and then using the quotient topology to define openness. This construction is simple to state but incredibly powerful.

From forming a circle by joining the ends of an interval to building a torus by gluing opposite sides of a square, identification spaces show how topology studies shape through relationships, not just through distances or measurements. Understanding them gives you a strong foundation for later topics in topological construction and classification.

Study Notes

An identification space is formed by taking a space $X$ and an equivalence relation $\sim$.
Points that are equivalent are glued together into one point in the quotient space $X/\sim$.
The natural map $q:X\to X/\sim$ sends each point to its equivalence class $[x]$.
The quotient topology says $U\subseteq X/\sim$ is open if and only if $q^{-1}(U)$ is open in $X$.
The quotient map $q$ is always surjective and continuous.
Identification spaces are a major tool in quotient topology.
Example: $[0,1]$ with $0\sim 1$ gives a space homeomorphic to a circle $S^1$.
Example: identifying opposite edges of a square gives a torus.
Equivalence relations must be reflexive, symmetric, and transitive.
Identification spaces help create and study new spaces by gluing parts together.