Embedding semigroups into semigroups with countability index two (Q1117334)

For semigroup S, define the countability index C(S) to be the least positive integer N such that every countable subset of S is contained in a semigroup with N generators, if such N exists; otherwise let \(C(S)=\infty\). For a topological space X, let S(X) denote the semigroup of continuous selfmaps. The main theorem here is that any semigroup S can be algebraically embedded in S(X) for some Hausdorff continuum X having \(C(S(X))=2\). In fact, X is a product of unit intervals.