Two topologies sit at the ends of the range. One declares every subset open, the other admits only the two the axioms force. Every other topology lies between them.

A camera with the finest grain records every speck; one with the coarsest sees a single block of colour. Both are pictures of the same scene, at the two ends of what can be told apart.

The discrete topology on $X$ is $\mathcal{P}(X)$, the collection of all subsets: everything is open. The indiscrete topology is $\{\varnothing,X\}$: only what the first axiom forces. If $\tau_1\subseteq\tau_2$ then $\tau_2$ is called finer and $\tau_1$ coarser, because the finer one can tell more sets apart.

• Both are genuinely topologies: every axiom is satisfied by each, and in the discrete case for trivial reasons.
• The discrete topology is the one chapter one’s 0-or-1 distance produces, since $B(x,1)=\{x\}$ makes every single point open.
• In the indiscrete topology the only open set holding a given point is the whole space, so no two points can be told apart by open sets.
• Finer and coarser only compare two topologies on the same set, and two topologies need not be comparable at all.

Suppose the indiscrete topology on a set with at least two points were the collection of open sets of some distance. Take two different points. Their distance is a positive number, so the ball around the first with that radius is an open set holding the first point and missing the second. It is neither empty nor the whole space, so it is an open set the indiscrete topology has not got. That is the contradiction, and it shows the axioms describe strictly more spaces than distances do.

The set has at least two points, so two different ones can be chosen.

Their distance cannot be zero: chapter one’s first condition says distance zero means the same point.

The centre is always in its own ball, and $y$ is not, because $d(x,y)$ is not less than $r$.

The ball is open, is not empty and is not everything, so the metric produces an open set the indiscrete topology does not contain.

All four subsets, all of them open.

The range within which every topology on $X$ falls.

The second point is exactly excluded, since the ball is strict.

Neither contains the other: not every pair can be compared.