The topology on a product is chosen to do one job: make the two projections continuous, and ask for nothing else. That means declaring the preimages of open sets under the projections to be open and taking the topology they generate — which is exactly what a subbasis is for.

A committee is given the smallest powers that let it do its two jobs and no more. Any further power would have to be justified, and nothing justifies it.

The subbasic open sets of a product: pull an open $U$ of one factor back through its projection. In the plane, $\pi_1^{-1}\big((a,b)\big)$ is the vertical strip over $(a,b)$ — open, unbounded upwards and downwards, and restricting one coordinate only.

The product topology on $X\times Y$ is the topology generated by the subbasis of all sets $\pi_1^{-1}(U)$ and $\pi_2^{-1}(V)$, with $U$ open in $X$ and $V$ open in $Y$. By the subbasis lesson it is the coarsest topology in which both projections are continuous.

• $\pi_1^{-1}(U)=U\times Y$ and $\pi_2^{-1}(V)=X\times V$: a subbasic set restricts one coordinate and leaves the other free.
• Intersecting two of them gives $U\times V$, so the boxes with open sides are a basis — the subject of the next lesson.
• Both projections are continuous by construction, and both are open maps: the image of a basic box under a projection is one of its sides.
• A projection is usually not a closed map. In the plane the hyperbola $\{(x,y):xy=1\}$ is closed, and its image under $\pi_1$ is $$, which is not.
• On $\mathbb{R}\times\mathbb{R}$ the product topology is the standard topology of the plane, which the lesson on metric products proves.

Continuity is by construction: the preimages of open sets were declared open. For openness, write an open set of the product as a union of basic boxes; the projection of a union is the union of the projections, and the projection of a box with open sides is one of those sides. So the image is a union of open sets of the factor. The argument uses only that images respect unions, which they do, unlike everything else images are asked to respect.

The preimages were made subbasic sets, so they are open and the projection is continuous.

Any open set of the product is a union of basic boxes.

Images respect unions, which is the one thing they do respect.

The projection of a non-empty box is its first side.

So the image is a union of open sets of the factor, and the projection is an open map.

The product topology is generated by the preimages of open sets under the projections: the smallest topology making both projections continuous.

• The product topology is generated by the preimages of open sets under the projections.
• Those preimages are the strips $U\times Y$ and $X\times V$.
• Both projections are continuous and open, and usually not closed.
• It is the coarsest topology making the projections continuous.

Great work! You now know how to:

• define the product topology by its subbasis;
• identify the strips as subbasic sets;
• use that the projections are continuous and open;
• say why the coarsest such topology is the one chosen.