A topology can be enormous — the open sets of the line are already uncountably many — and yet be described by a countable list. Second countability asks for a countable basis for the whole space, first countability only for a countable basis at each point. Both hold in the spaces this course is built on, and the second is what makes sequences enough to detect closures and continuity.

A language has infinitely many sentences and a finite dictionary. Asking for a countable basis is asking for a dictionary; asking for a countable basis at a point is asking for one per speaker.

A countable basis at a point of the line: the balls $B(x,1/n)$. Every open set around $x$ contains one of them, so this countable list describes every neighbourhood of $x$ — which is first countability. Second countability asks for one countable list serving the whole space at once, and the intervals with rational endpoints are such a list.

A space is first countable when every point has a countable family of open sets around it such that every open set around the point contains one of them. It is second countable when the topology has a countable basis. Second countable implies first countable, and the converse fails.

• Every metric space is first countable: the balls $B(x,1/n)$ are a countable basis at $x$, since any open set around $x$ holds a ball and any ball holds one of these.
• $\mathbb{R}$ is second countable: the intervals with rational endpoints are a basis, and there are countably many of them. ${\mathbb{R}}^{}$ is too, by taking boxes with rational corners, which is why the plane behaves as well as the line.

Take a countable basis for the whole space and a point. Keep the basic sets that contain the point and discard the rest. What is left is a subfamily of a countable family, so it is countable. It is also a basis at the point: any open set around the point is a union of basic sets, and the point lies in one of them, which therefore contains the point and sits inside the open set. So the point has a countable basis around it, which is first countability. Nothing in the argument is about a particular point, so it holds at every point at once.

Start from a countable basis for the whole topology.

Keep the basic sets containing the chosen point.

A subfamily of a countable family is countable.

Every open set around the point is a union of basic sets, and one of them catches the point.

The kept family is a countable basis at the point, and the point was arbitrary.

Second countability asks for a countable basis for the topology; first countability only for one at each point.

• Second countable: a countable basis for the topology. First countable: one at each point.
• Second implies first; metric spaces are first countable and the line is second countable.
• Separable means a countable dense subset, and second countable implies separable.
• The Sorgenfrey line is first countable, separable and not second countable.

Great work! You now know how to:

• tell the two countability axioms apart;
• produce a countable basis for the line and for the plane;
• see why metric spaces are first countable;
• name the space that is first countable and not second.