Topological hyperplanes are like hyperplanes in Rn except that they need not be straight. The properties they do have are that they separate the whole space into two parts and the intersection of any two of them, if not empty, is a topological hyperplane in each.
An arrangement of topological hyperplanes is a finite set; a region of the arrangement is a component of the complement. We ask about the number and topology of the regions and how arrangements of topological hyperplanes differ from those of true hyperplanes.
This is joint work with the well-remembered David Forge.