Finite products and projections

Try

A point of X×Y contains two coordinates. To control it locally, choose an open neighborhood U of its first coordinate and V of its second, then take the rectangle U×V. In R² these need not be squares or disks; rectangles generate the same usual topology.

A product topology records independent coordinate tests. It does not mean every open set itself is a single rectangle. A disk, for example, is a union of many rectangles lying inside it.

Predict

Predict whether coordinate projections are continuous. Requiring the first coordinate to lie in U gives the inverse image U×Y, which is open.

Predict whether projections must send closed sets to closed sets. The hyperbola xy=1 is closed in R², but its projection onto the first coordinate omits only zero and is not closed. This separates continuity, openness and closedness of a map.

Experiment

Use rectangle calculations rather than a picture alone.

Operation	Result
First-coordinate inverse image of U	U×Y
Second-coordinate inverse image of V	X×V
Intersection of these two sets	U×V
Projection of U×V when V is nonempty	U

If V is empty, the last image is empty. Keeping that case explicit makes the open-projection proof valid even for empty factors.

Understand

Open rectangles form a basis: they cover X×Y, and the intersection of two rectangles is the rectangle formed by intersecting their respective coordinates. The resulting topology is the product topology.

A map f:Z→X×Y is continuous if and only if both coordinate functions are continuous. Necessity follows by composition with the continuous projections. Conversely, the inverse image of U×V is the intersection of the two coordinate preimages, hence open. Arbitrary unions extend this from basic rectangles to every product-open set. This is the two-coordinate version of the initial-topology criterion.

Each projection is also an open map. An open set is a union of rectangles; each projected rectangle is open or empty, and projection preserves unions. This proof does not make the projection closed, as the hyperbola example shows.

For a fixed y in a nonempty factor, x↦(x,y) is continuous by the coordinate criterion, and the first projection restricts to its continuous inverse on the slice with second coordinate y. Thus each slice is homeomorphic to its factor with the subspace topology. No assumption that the singleton is open is needed.

For a finite product of more factors, repeat the same argument. Basic neighborhoods restrict each coordinate to an open set, and finitely many coordinate preimages can be intersected. The finiteness of this intersection will become decisive in the infinite-product lesson.

Apply

Prove the hyperbola H is closed using the continuous multiplication map m(x,y)=xy and the closed singleton containing one. Then compute its projection exactly. A drawing that approaches the y-axis without touching it can suggest the answer but does not prove closedness or the complete image.

Repair “A product-open set must project to a closed set because the coordinate is forgotten.” Forgetting information has no such implication. Use the established open-projection theorem and a separate argument for any proposed closedness claim.

Continue

Infinite products retain finitely many simultaneous coordinate restrictions, even when there are infinitely many coordinates available. Optional depth: show that a product of two Hausdorff spaces is Hausdorff by finding one coordinate in which two distinct points differ and pulling back disjoint neighborhoods from that coordinate.