Let \(S\) be a semigroup and \(U\) be a subset of \(S\). The relative rank of \(S\) with respect to \(U\) is the minimal cardinality of a subset \(V\) of \(S\) such that \(U\) together with \(V\) generate \(S\). The principal result in the theory of relative ranks of semigroups of mappings, where the composition of mappings is the semigroup operation, is due to W. Sierpiński [Fundam. Math. 24, 209–212 (1935; Zbl 0011.10607)]. It says that the relative rank of the semigroup of all mappings \(A^A\) from an infinite set \(A\) to \(A\) with respect to any subsemigroup is either uncountable or finite and then equal to 0, 1 or 2. Note that S. Banach in [Fundam. Math. 25, 5–6 (1935; Zbl 0011.25305)] gave a very nice short proof of this theorem.
In the present paper, the authors consider a relative rank that depends only on topology, namely the relative rank of the semigroup of all Borel measurable mappings with respect to the semigroup of continuous mappings for some classical topological spaces. In their main result, they show that this rank is equal to the first uncountable cardinal \(\aleph_1\) for a wide family of Polishable topological spaces \(X\), namely those with either can be retracted to a Cantor subset of \(X\), so, in particular, all uncountable zero-dimensional spaces, or contain a topological copy of the interval \([0,1]\) (so, in particular, Euclidean spaces), or homeomorphic to their Cartesian square \(X^2\).