One set can carry many topologies, and they are compared by containment: more open sets means finer. Doing this by listing is hopeless for the line, so the comparison is made through bases — and the test is the point test again, run between two families.

Two maps of the same country, one with more detail than the other. The finer map shows everything the coarser one does; whether it is finer is settled by checking, around each place, that it has a smaller region than the coarse map does.

Why the lower-limit topology is finer than the standard one: inside any interval around $x$ there is a half-open interval $[x,b)$ around $x$. The test runs one way only — there is no interval around $0$ inside $[0,1)$ — and that is what makes the containment strict.

Two topologies on one set are compared by inclusion: $\tau'$ is finer than $\tau$ when $\tau\subseteq\tau'$, and $\tau$ is then coarser. When both are given by bases the comparison can be made one point at a time, which is the criterion below. Topologies on a set need not be comparable at all.

• The discrete topology is the finest on any set and the indiscrete one the coarsest, so every topology lies between them.
• $\mathbb{R}_\ell$ is strictly finer than the standard topology, and the standard topology is strictly finer than the cofinite topology on $\mathbb{R}$.
• The intersection of any family of topologies on $X$ is again a topology. The union of two topologies usually is not: it can fail the intersection axiom.
• Finer means more open sets, so it means more continuous maps out of the space and fewer into it — a trade chapter 4 makes precise.
• Two topologies with a common basis are equal. That is how the Euclidean, taxicab and maximum metrics were shown to give one topology on $\mathbb{R}^n$.

If the finer topology contains the coarser one, a basic set of the coarse topology is open in the fine one, so the point test for the fine basis supplies the smaller set. Conversely, take any open set of the coarse topology and a point in it: a coarse basic set sits around the point inside, and the condition puts a fine basic set inside that. So the coarse open set satisfies the point test for the fine basis and is open in the fine topology.

A basic set of the coarser topology is open in the finer one.

The point test for the finer basis then gives the smaller set, which is the condition.

For the converse start with an open set of the coarser topology and use its own basis.

The condition supplies a finer basic set inside, around the same point.

Every point of the set has a finer basic set around it inside, so the set is open in the finer topology.

$\tau'$ is finer than $\tau$ when it contains it as a collection. The finer topology can tell more sets apart.

• $\tau'$ finer than $\tau$ means $\tau\subseteq\tau'$: more open sets.
• Through bases: inside every coarse basic set around a point there must be a fine one around it.
• $\mathbb{R}_\ell$ is finer than the standard topology, which is finer than the cofinite one.
• Intersections of topologies are topologies; unions need not be, and two topologies may be incomparable.

Great work! You now know how to:

• say what finer and coarser mean;
• compare two topologies through their bases;
• place the lower-limit, standard, cofinite and discrete topologies in order;
• give two topologies that cannot be compared at all.