Intersecting two subbasic strips, one from each factor, gives an open box. The boxes are a basis for the product topology, which makes the product concrete: an open set of the plane is a union of open rectangles, exactly as it is a union of open discs.

Two rulings across a sheet, one vertical and one horizontal, cut it into rectangles. Any shape drawn on the sheet is approximated from inside by rectangles, and in the end built from them.

The basic open sets of the plane as a product: boxes $(a,b)\times(c,d)$ with open sides, drawn dashed because the edges do not belong. Around any point of any open set one of them fits, which is the point test — and it is why the product topology on $\mathbb{R}\times\mathbb{R}$ is the topology of the plane.

The sets $U\times V$ with $U$ open in $X$ and $V$ open in $Y$ form a basis for the product topology. It is enough to take $U$ and $V$ from bases of the factors, which for the plane gives the open rectangles with rational corners — a countable basis.

• $U\times V=\pi_1^{-1}(U)\cap\pi_2^{-1}(V)$, so a box is a finite intersection of subbasic sets, which is what the subbasis lesson said a basic set would be.
• The two conditions for generating a topology hold: the boxes cover, and an intersection of two boxes is a box.
• If $\mathcal{B}_X$ and $\mathcal{B}_Y$ are bases for the factors, the boxes $B\times C$ from them form a basis for the product. In the plane: rectangles with rational corners.
• A union of boxes is almost never a box, which is why the boxes are a basis and not the topology.
• For finitely many factors the same works with $U_1\times\cdots\times U_n$. For infinitely many, a basic set restricts only finitely many coordinates, and that is where the product topology parts company with the box topology.

The product topology is generated by the strips, so a basic set of it is a finite intersection of strips. Grouping the ones coming from the first factor and the ones from the second, and intersecting each group, leaves one strip from each side, and their intersection is a box. So every basic set is a box, and the point test for the boxes follows from the point test for the basis the subbasis produced.

A subbasis generates a basis of finite intersections, and one of those sits between the point and the open set.

Each factor of the intersection comes from one projection or the other.

Preimages respect intersections, so each group collapses to a single strip with an open side.

Two strips, one from each factor, meet in a box with open sides.

So a box sits between the point and the open set, which is the point test for the boxes.

A basic open set of a product is an intersection of one strip from each factor, which is a box with open sides.

• The boxes $U\times V$ with open sides are a basis for the product topology.
• Bases of the factors give a basis of the product, so countable bases multiply.
• Open sets of a product are unions of boxes and are usually not boxes.
• In an infinite product a basic set restricts finitely many coordinates.

Great work! You now know how to:

• use the boxes with open sides as a basis;
• build a basis of a product from bases of the factors;
• apply the point test in a product;
• say what a basic set of an infinite product restricts.