The main result of the paper is that for every \(n\in \mathbb{Z}_+\) and every function \(d:{\mathcal B}_n\to \{0,1,\dots,\infty\}\), where \({\mathcal B}_n\) is the class of all nonempty subsets of \(\{0,1,\dots,n\}\), there exist a space \(X=N\cup{\mathcal R}\) and a decomposition \(N=N_1\cup\dots\cup N_n\) so that letting \(X_i=\overline{N}_i\) we have, for every \(A\in {\mathcal B}_n\), \(\dim\bigcup \{X_i: i\in A\}=d(A)\). This also demonstrates the failure of the monotonicity of the covering dimension.
Reviewer’s remark: In a discussion of the history of \(N\cup {\mathcal R}\) the author points out that L. Gillman and M. Jerison made a wrong attribution in their book [Rings of continuous functions (1960; Zbl 0093.30001)]. The correct reference is [S. Mrówka, On completely regular spaces, Fundam. Math. 41, 105-106 (1954; Zbl 0055.41304)].