A topological space \(X\) is countable dense homogeneous (CDH) provided that \(X\) is separable and that if \(A\) and \(B\) are countable dense subsets of \(X\) then there is an autohomeomorphism \(h\) on \(X\) such that \(h(A)= B\). Also \(X\) is strongly locally homogeneous (SLH) provided that \(X\) has a base of open sets \(U\) such that if \(x,y\in U\) then there is an autohomeomorphism on \(X\) that is supported on \(U\) and that takes \(x\) to \(y\). This paper gives a long detailed argument showing the existence (assuming the continuum hypothesis) of a connected CDH subspace of the plane that is not SLH. This space is the union of a transfinite sequence of subsets of the plane constructed by three levels of induction. The resulting space is not SLH because it is a connected dense subset of the plane that intersects vertical lines only once.