Compact group topologies for 𝑅

There is a compact solenoidal group S that is group isomorphic to the additive real numbers R . The existence of S leads to a study of compact real groups; that is, compact groups which are group isomorphic to R . The compact real groups are characterized as products of S . Various conditions on a topological group G are given which are equivalent to G being compact real. It is shown that any compact real group is solenoidal, connected and separable. Many sets and functions on R which are measurable with respect to the usual topology are shown to be nonmeasurable with respect to any compact group topology.