Are there useful topologies on Cartesian products “in between” the product topology and the box topology?
On the Cartesian product of topological spaces, there are two standard topologies: One is the product topology, the other is the box topology.
As is well known, the box topology is generated by the product of open sets, and the product topology is generated by such products with the restriction that only finitely many factors are not the full space.
However in principle there could be other topologies defined on the product which sit “in between” product and box topology. As a simple example, one might demand that at most countably many factors are not the full space. Or one might take the index set to have a topology, and demand that the set of indices corresponding to non-full spaces is relatively compact.
My question now is: Are there any such ”in between” topologies that are actually known to be useful?
1 answer
Disclaimer: this is not a complete answer. Indeed, the OP has likely thought of some or all of the possibilities mentioned. I answer in the hope that it will be helpful to other users and perhaps even spur them on to think of a more complete answer.
In the special case where all the factors in a product of spaces are the same, there are two topologies I know of that are commonly used:
- The topology of uniform convergence, wherein a sequence/filter of functions $f_n$ converges to $f$ iff it converges uniformly. This is the natural topology for sequences or functions in whose uniform convergence we are interested; the box topology, by contrast is the topology of pointwise convergence.
- The compact-open topology, which is generally used for spaces of continuous functions $\mathcal{C}(\text{Dom}, \text{Cod})$. It is weaker than the uniform topology and I think strictly weaker iff the domain $\text{Dom}$ of the functions is not compact. The compact-open topology captures the intuition that we ought to be able not only to approximate particular values of a continuous function (as we can in the product topology) but be able to obtain ever more precise bounds of its value over "reasonable" (namely compact) regions.
In the case where the factors are not necessarily the same, the topology of uniform convergence seems to be a pretty natural way of taking the product of metric topologies, though I don't know off the top of my head where this concept has been used.
This topology can also be generalized to a product of arbitrary subsets of a uniform topological space: just take as one's basis the product of open sets taken from some uniform cover. This reduces to the topology of uniform convergence if the uniform topological space is a metric space.
As for the OP's idea of "demand[ing] that at most countably many factors are not the full space", I admit I couldn't figure out much to do with it. I thought that maybe one could prove something analogous to the Alexander subbase lemma and then derive a theorem that the product of Lindelöf spaces, under this topology, is Lindelöf, but I'm pretty sure that won't work.
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