On a Conjecture of Koch1

Let A" be a topological space. We recall that D, a subset of X is called a C-set if any continuum which meets D and its complement must contain D. Let 5 be a continuum which is a topological semigroup with identity 1, and let H denote the maximal subgroup of 5 containing 1. It is well known that H exists and is compact. The following four conjectures have been raised and shown to be equivalent by Koch, [2V (1) The unit is not a weak-cutpoint. (2) S is aposyndetic at any point with respect to 1. (3) The identity component of H is not a nontrivial C-set. (4) 1 belongs to no nontrivial C-set. We give here an affirmative answer to these conjectures. (We assume, of course, that 5 is not a group.)