Infinite products and the box topology

Try

An infinite real sequence can be viewed as a point of R^N. To specify a product neighborhood, one might require its first coordinate to be near zero and its fifth coordinate to be near two, leaving all other coordinates unrestricted.

The box topology allows a separate restriction in every coordinate at once. Both constructions sound natural, but they lead to different open sets and different convergence behavior. The word product conventionally refers to the finite-restriction topology.

Predict

Predict whether the points x^(n)=(1/n,1/n,…) approach the zero sequence. Every coordinate does, so they converge in the product topology.

Now impose the box neighborhood whose jth coordinate lies in (−1/j,1/j). For every n, choose a coordinate j at least n. The value 1/n does not lie in that strict interval. This single box excludes the entire sequence.

Experiment

Compare neighborhood requirements.

Topology on R^N	Allowed basic restriction
Product	Open restrictions in finitely many coordinates
Box	Open restrictions in every coordinate
Product example	First coordinate in (−1,1), others free
Box example	jth coordinate in (−1/j,1/j) for every j

The box example is not product-open at zero because any finite-coordinate neighborhood leaves some coordinate free to violate its shrinking bound.

Understand

For spaces X_i indexed by a set I, the product topology is generated by the subbasic sets π_i⁻¹(U), where π_i is a coordinate projection and U is open in X_i. Finite intersections produce basic cylinders with finitely many restricted coordinates. Every projection is continuous, and this is the coarsest topology with that property.

The initial-topology criterion proves h:Z→∏X_i continuous exactly when all coordinate maps are continuous. The sufficiency argument checks subbasic preimages and then uses preimage preservation of finite intersections and unions. An infinite intersection is not required to be open.

Convergence is coordinatewise as well. If a net converges in the product, continuity of each projection gives coordinate convergence. Conversely, a basic neighborhood imposes finitely many coordinate conditions. Each holds beyond some index; directedness supplies a common later index for all finitely many thresholds. For sequences this is simply their maximum. Thus the net eventually lies in the neighborhood.

The box topology has products of arbitrary open coordinate sets as a basis. It is finer than the product topology, because a basic cylinder is a box with most factors equal to their whole spaces. On R^N it is strictly finer by the shrinking-box example. Coordinatewise continuity or convergence need not control infinitely many simultaneous restrictions, so the corresponding product theorems cannot automatically be transferred to the box topology.

Apply

Prove that the diagonal map d:R→R^N defined by d(t)=(t,t,…) is product-continuous: every coordinate is the usual identity. Its inverse image of the shrinking box is the singleton zero, which is not usual-open, so the same map is not box-continuous.

Repair “Checking each coordinate separately checks them all at once.” Separate checks can have thresholds depending on the coordinate. Product neighborhoods require only finitely many thresholds at a time; box neighborhoods may demand a common threshold across infinitely many coordinates, which need not exist.

Continue

Later, compactness of arbitrary products will be proved using ultrafilters and an explicit choice principle. That theorem applies to the product topology. Optional depth: derive a metric for a countable product of bounded metric spaces using a weighted sum, and explain why it produces finite-coordinate neighborhood behavior rather than the box topology.