Chapter one proved three things about the open sets of a metric space. Those three facts now become the definition, and the distance that produced them is thrown away.

Keep the rules a game is played by and forget the town they were written in. Anyone following the same rules is playing the same game.

A topology on a set $X$ is a collection $\tau$ of subsets of $X$ obeying three conditions. Its members are called the open sets, and the pair is a topological space. Nothing in the conditions mentions a distance, a ball or a radius. Write $\tau_d$ for the collection of sets chapter one called open in a metric space.

• The first condition is the floor. Without it the collection could be empty and nothing would follow.
• Unions carry no restriction on how many; intersections are restricted to finitely many. That asymmetry was a theorem in chapter one and is an assumption here.
• Being open is no longer a property a set has on its own. It is membership of $\tau$, and a different $\tau$ gives a different answer.
• One set carries many topologies, so a subset open in one may fail to be open in another.

Chapter one checked all three conditions already, one lesson at a time. The empty set and the whole space were shown open directly. A union of open sets was shown open with no restriction on size. An intersection of finitely many was shown open by taking the smallest of finitely many positive radii. Nothing is left to prove, because the axioms were chosen to be exactly those three facts — which is the point: the new definition loses nothing.

The empty set has no point at which the ball condition could fail, and every ball lies inside the whole space.

A point of a union lies in some member, that member hands over a ball, and a ball inside one member is inside the union.

Each member supplies a radius at the point, and the smallest of finitely many positive radii is still positive.

The three axioms are precisely the three facts just recalled, so the metric-open sets form a topology.

The first condition, which chapter one proved directly.

One subset, two collections, two answers.

Every interval open, the intersection a single point.

The three checks were done, one per lesson, in chapter one.