The Box Product Problem

Scott W. Williams

Given infinitly (countably) many copies of the unit interval, the typical (Tychonov) product topology on their product is topologically a copy of the Hilbert Cube. Give it the box product topology instead (so open sets are unions of productts of open intervals). Is it normal ;i.e., van disjoint closed sets be separated by disjoint open sets? The anwer is no for uncountably many copies, but the continuum hypothesis implies yes for the countable case.

Box Topology: The topology of the line is generated by open intervals; i.e., (unions of) sets of form (a,b). The topology of the plane, R^xR=R2, is generated by "open" squares; i.e, sets of form

(a1,b1)^x(a2,b2).

The topology of 3-space, R3, is generated by "open" boxes; i.e, sets of form

(a1,b1)^x(a2,b2)^x(a3,b3).

The box topology on the set R, is generated by "open" boxes; i.e, sets of form

(a1,b1)^x(a2,b2)^{x ... x}(an,bn)^{x ...}.

The box topolgy is defined on all infinite products similarly, and a box product is the product, as sets, of topological spaces given the box topology.

Normal: A space X is normal provided for each pair A,B of disjoint closed sets