On problems of H. Cook (Q1088186)

Let \({\mathcal F}\) be a class of mapping such that all homeomorphisms are in \({\mathcal F}\) and the composition of any two mappings in \({\mathcal F}\) is also in \({\mathcal F}\). In University of Houston Mathematics Problem Book (Problem 150 and Problem 151), H. Cook has the following problems: (i) If a continuum X is homogeneous with respect to \({\mathcal F}\), is there a continuum Y which is \({\mathcal F}\)-equivalent to X and which is homogeneous? (ii) Suppose that a continuum Y is \({\mathcal F}\)-equivalent to a homogeneous continuum and \(\epsilon\) is a positive number. Does there exist a positive number \(\delta\) such that if a,b\(\in Y\) and \(d(a,b)<\delta\), then there is a mapping f in \({\mathcal F}\) from Y onto Y such that \(f(a)=b\) and no point of Y is moved a distance more than \(\epsilon\) ? We give negative answers to the problems (i) and (ii).