The line and the plane carry one topology so familiar that it is worth pinning down: the sets built from open intervals and open discs. Everything later in the course is tested against it, and it has a description that will become the idea of a basis — every open set is a union of the simple ones.

A language with a small vocabulary can still say a great deal, because sentences are built by joining words. Open intervals are the vocabulary of the line, and unions are the only grammar there is.

An open subset of the line: a union of open intervals, drawn with hollow ends because the endpoints are missing. Every open set of $\mathbb{R}$ looks like this, and the union may be of infinitely many pieces — which is how a set like $\mathbb{R}\setminus\{0\}$, or the complement of the whole of $\mathbb{Z}$, is open.

The standard topology on $\mathbb{R}$ is the one chapter 1 produced from the metric $|x-y|$: a set $U$ is open when each of its points has an interval around it inside $U$. On $\mathbb{R}^2$ the same definition with discs — or with squares, which give the same answer — produces the standard topology of the plane. Being open is membership of this collection, so the question is always open in which topology.

• $(0,1)$ is open and $[0,1]$ is not: no interval around $0$ stays inside the closed one. $[0,1)$ is open at no end that matters — it fails at — so it is not open either.

One direction collects the intervals the definition already provides: each point of an open set has one inside the set, and the union of all of them is the set itself, since every point belongs to its own interval and no interval leaves the set. The other direction is the union axiom: an open interval is open, and a union of open sets is open. So the open sets of the line are exactly what unions of intervals can reach.

Openness hands each point an interval around it inside the set.

Every point lies in its own interval, so the union catches all of them.

Each of those intervals was chosen inside the set, so their union cannot leave it.

Conversely an interval is open and the union axiom makes any union of open sets open.

A subset of the line is open when each of its points has an interval around it inside the set. One bad point is enough to spoil it.

• The standard topology on the line: open means every point has an interval around it inside.
• Every open subset of the line is a union of open intervals, possibly infinitely many.
• $\mathbb{Q}$, a point and a line in the plane are not open; open discs, squares and unions of intervals are.
• Different metrics can produce the same topology.

Great work! You now know how to:

• decide whether a subset of the line or plane is open;
• write an open set as a union of open intervals;
• name sets that are open in one topology and not in another;
• see why several metrics can give one topology.