Comparing Topologies

In this lesson, students, you will learn how to compare topologies on the same set and decide when one topology is finer, coarser, or equal to another 🌍. This is a key idea in topology because many problems ask not just what a topology is, but how two topologies relate to each other. By the end of this lesson, you should be able to explain the vocabulary, test whether one topology contains another, and connect these ideas to bases and subbases.

Objectives:

Explain the main ideas and terminology behind comparing topologies.
Apply topology reasoning to decide whether one topology is finer, coarser, or equal.
Connect comparing topologies to bases and generated topologies.
Use examples to support conclusions about topological comparison.

Think of comparing topologies like comparing ways of organizing the same set of objects. A topology tells you which collections are considered open. If one topology allows more open sets than another, it gives you more flexibility. This difference matters in continuity, convergence, and many other parts of mathematics 📘.

What it means to compare topologies

Suppose $X$ is a set and $\mathcal{T}_1$ and $\mathcal{T}_2$ are topologies on $X$. We compare them by looking at their open sets.

$\mathcal{T}_1$ is finer than $\mathcal{T}_2$ if $\mathcal{T}_2 \subseteq \mathcal{T}_1$.
$\mathcal{T}_2$ is coarser than $\mathcal{T}_1$ if $\mathcal{T}_2 \subseteq \mathcal{T}_1$.
The two topologies are equal if $\mathcal{T}_1 = \mathcal{T}_2$.
If neither topology is contained in the other, they are incomparable.

The word “finer” means “has more open sets,” while “coarser” means “has fewer open sets.” So if $\mathcal{T}_1$ is finer than $\mathcal{T}_2$, then every open set in $\mathcal{T}_2$ is also open in $\mathcal{T}_1$.

A helpful way to remember this is to imagine two nets. A finer net has smaller holes, so it catches more detail. In the same way, a finer topology has more open sets and more detailed structure 🕸️.

For example, on a set $X$, the discrete topology $\mathcal{P}(X)$ is the finest topology because every subset is open. The indiscrete topology $\{\varnothing, X\}$ is the coarsest topology because it has the fewest open sets possible.

How to decide whether one topology is finer than another

To show that $\mathcal{T}_1$ is finer than $\mathcal{T}_2$, you must prove that every open set in $\mathcal{T}_2$ belongs to $\mathcal{T}_1$.

This is a subset test:

$$\mathcal{T}_2 \subseteq \mathcal{T}_1.$$

Sometimes this is easy if the topologies are listed directly. For instance, let $X = \{a,b,c\}$ and

$$\mathcal{T}_1 = \{\varnothing, \{a\}, \{a,b\}, X\}$$

$$\mathcal{T}_2 = \{\varnothing, \{a\}, X\}.$$

Then $\mathcal{T}_2 \subseteq \mathcal{T}_1$, so $\mathcal{T}_1$ is finer than $\mathcal{T}_2$.

But in many problems, the topologies are not given as complete lists. Instead, they may be described by bases. In that case, the same idea still applies, but you may need to use the basis properties to check whether every basis-generated open set in one topology is open in the other.

Here is a practical strategy:

Identify a basis or subbasis for each topology.
Check whether each basic open set of the coarser topology is open in the finer one.
If yes, then the topology generated by that basis is contained in the other topology.

If you can show that every basis element of $\mathcal{B}_2$ is open in $\mathcal{T}_1$, then because every open set in the topology generated by $\mathcal{B}_2$ is a union of basis elements, it follows that the whole topology is contained in $\mathcal{T}_1$.

Examples of comparing familiar topologies

A classic example is the real line $\mathbb{R}$.

Example 1: Standard topology vs. lower limit topology

The standard topology on $\mathbb{R}$ has basis sets of the form $$(a,b).$$

The lower limit topology has basis sets of the form $$[a,b).$$

Which one is finer?

Take any standard open interval $(a,b)$. It can be written as a union of lower limit basis sets:

$$(a,b) = \bigcup_{x \in (a,b)} [x,b).$$

Each set $[x,b)$ is open in the lower limit topology, so $(a,b)$ is also open there. Since every standard open interval is open in the lower limit topology, every standard open set is open in the lower limit topology.

Therefore, the lower limit topology is finer than the standard topology.

This is a great example of how one topology can contain another while still living on the same set. It also shows why “more open sets” means “finer” ✅.

Example 2: Discrete and indiscrete topologies

Let $X$ be any set.

The discrete topology is $\mathcal{P}(X)$.
The indiscrete topology is $\{\varnothing, X\}$.

Since $\{\varnothing, X\} \subseteq \mathcal{P}(X)$, the discrete topology is finer than every topology on $X$, and the indiscrete topology is coarser than every topology on $X$.

These two topologies are important extremes. They help you check whether your reasoning is sensible. If a topology is supposed to be finer than all others, it must at least contain every open set from any other topology.

Example 3: Subspace topologies

If $Y \subseteq X$ and $X$ has topology $\mathcal{T}$, the subspace topology on $Y$ is

$$\mathcal{T}_Y = \{U \cap Y : U \in \mathcal{T}\}.$$

This topology is usually compared with other topologies on $Y$. If another topology on $Y$ has more open sets than $\mathcal{T}_Y$, then it is finer. If it has fewer, it is coarser.

A common mistake is to compare sets of open sets without checking that the underlying set is the same. Topologies are only compared when they are on the same set $X$ or the same subset $Y$.

Bases, subbases, and generated topologies

Comparing topologies is closely connected to bases and subbases because many topologies are defined by generators.

A basis $\mathcal{B}$ for a topology on $X$ is a collection of sets such that every open set is a union of basis elements. The topology generated by $\mathcal{B}$ is the collection of all unions of elements of $\mathcal{B}$.

A subbasis is a collection of sets whose finite intersections form a basis. This means a topology can be built in two stages:

Start with a subbasis.
Take finite intersections to get a basis.
Take arbitrary unions to get the generated topology.

When comparing topologies generated this way, you often compare their generating collections.

Here is a useful fact:

If every basis element of $\mathcal{B}_2$ is open in $\mathcal{T}_1$, then the topology generated by $\mathcal{B}_2$ is contained in $\mathcal{T}_1$.
If every subbasic set of a subbasis $\mathcal{S}_2$ is open in $\mathcal{T}_1$, then the topology generated by $\mathcal{S}_2$ is also contained in $\mathcal{T}_1$.

Why? Because the topology generated by the subbasis is built from unions and finite intersections of those sets, and topologies are closed under finite intersections and arbitrary unions.

This is a powerful method for proving one topology is finer than another without listing every open set.

Real-world style intuition and careful reasoning

Think about a social network app 📱. One way to group users might be very detailed: by school, grade, and club. Another way might be broader: only by school. The more detailed system is like a finer topology because it creates more possible open groups.

Mathematically, if a topology has many open sets, it gives you more ways to describe local behavior. This affects continuity. A function $f : X \to Y$ is continuous when the preimage of every open set in $Y$ is open in $X$. If the topology on $Y$ becomes finer, there are more open sets to check, so continuity becomes harder to satisfy.

That is why comparing topologies matters. A finer topology usually makes continuity stricter, while a coarser topology usually makes continuity easier.

For example, if $\mathcal{T}_1$ is finer than $\mathcal{T}_2$ on $X$, then the identity map

$$\operatorname{id} : (X, \mathcal{T}_1) \to (X, \mathcal{T}_2)$$

is continuous, because every open set in $\mathcal{T}_2$ is already open in $\mathcal{T}_1$.

This simple fact is one of the main reasons comparison is so useful.

Conclusion

Comparing topologies means determining how their open sets relate. If $\mathcal{T}_1 \supseteq \mathcal{T}_2$, then $\mathcal{T}_1$ is finer and $\mathcal{T}_2$ is coarser. If the topologies are equal, they have exactly the same open sets. If neither contains the other, they are incomparable.

This topic fits naturally into bases and generated topologies because many comparisons are easiest when you work from bases or subbases instead of full topologies. By checking whether generating sets are open in another topology, you can prove containment and understand how topological structure changes. students, this is a core tool for reading and building mathematical arguments in topology 📚.

Study Notes

A topology $\mathcal{T}_1$ is finer than $\mathcal{T}_2$ if $\mathcal{T}_2 \subseteq \mathcal{T}_1$.
A topology $\mathcal{T}_2$ is coarser than $\mathcal{T}_1$ if $\mathcal{T}_2 \subseteq \mathcal{T}_1$.
Two topologies are equal if $\mathcal{T}_1 = \mathcal{T}_2$.
If neither topology is contained in the other, they are incomparable.
The discrete topology $\mathcal{P}(X)$ is the finest topology on $X$.
The indiscrete topology $\{\varnothing, X\}$ is the coarsest topology on $X$.
To compare topologies, check whether every open set of one is open in the other.
If a basis element of one topology is open in another topology, that helps prove containment.
If every subbasic set of one topology is open in another topology, then the generated topology is contained in the other topology.
Comparing topologies is important for continuity, since a finer codomain topology makes continuity harder to satisfy.