Listing every open set of the line is hopeless; describing them all as unions of intervals takes one sentence. A family that does this for a topology is called a basis, and most topologies are met through one rather than through the collection itself.

A paint catalogue does not list every colour. It lists a few pigments and says that everything else is a mixture, which is shorter and loses nothing.

The test a basis has to pass: for every open $U$ and every point $x$ in it, some member $B$ of the basis satisfies $x\in B\subseteq U$. Collecting one such $B$ per point and taking the union rebuilds $U$ exactly, which is why a basis carries the whole topology.

Let $(X,\tau)$ be a topological space. A collection $\mathcal{B}\subseteq\tau$ of open sets is a basis for $\tau$ when every open set is a union of members of $\mathcal{B}$. The members are often called basic open sets. A topology has many bases, and a small one is a short description of the whole space.

• The open intervals are a basis for the standard topology on $\mathbb{R}$: that was the theorem of the lesson on open sets.
• So are the intervals with rational endpoints, and there are only countably many of those — a fact chapter 8 turns into second countability.
• In $\mathbb{R}^2$ the open discs are a basis, and so are the open rectangles $(a,b)\times(c,d)$. Two different bases, one topology.
• In a discrete space the singletons $\{x\}$ form a basis, and no smaller family will do: each singleton is open and is a union only of itself.
• The whole topology is always a basis for itself, so every topology has one. The interest is in finding a small basis, not in having any.

If every open set is a union of basic sets, a point of an open set lies in one of the pieces of that union, and the piece lies inside the set. Conversely, if each point of an open set has a basic set around it inside, collect one for each point: their union contains every point of the set, and none of them leaves the set, so the union is the set itself. The two conditions are the same statement counted in two ways.

A point of a union lies in one of the pieces, and every piece lies inside the set.

For the converse, use the point test once for each point of the set.

Each point lies in its own chosen set, so the union catches all of them.

Each chosen set lies inside the open set, so their union cannot leave it.

The two inclusions make the set a union of basic sets, which is what a basis has to provide.

A basis is a family of open sets from which every open set can be built by taking unions.

• A basis is a family of open sets whose unions give every open set.
• Point test: for every open $U$ and $x\in U$ there is a basic $B$ with $x\in B\subseteq U$.
• Intervals, and intervals with rational endpoints, are bases for the line; discs and rectangles for the plane.
• A topology has many bases; the singletons are the smallest basis of a discrete space.

Great work! You now know how to:

• say what a basis for a topology is;
• apply the point test to decide whether a family is one;
• name bases for the line, the plane and a discrete space;
• see that one topology has many bases.