On compactifications preserving the dimension of spaces (Q1612221)

For a \(T_4\) space \(X\), \(\dim X\) is the covering dimension of \(X\), \(w(X)\) the weight of \(X\), and if \(Y\) and \(Z\) are compactifications of \(X\), then we write \(Y\leq Z\) when there is a continuous map \(Z\to Y\) which is the identity on \(X\). It is known [\textit{E. G. Skljarenko}, On embedding of normal spaces in bicompacta of the same weight and dimension, Dokl. Akad. Nauk SSSR 123, No. 1, 36-39 (1958; Zbl 0089.38901)] that every \(T_4\) space has a compactification \(Y\) with \(w(Y)= w(X)\) and \(\dim Y=\dim X\). The authors sharpen this result by proving that if \(X\) is a \(T_4\) space then for every compactification \(Y\) of \(X\) there is a compactification \(Z\) of \(X\) such that \(w(Z)= w(X)\), \(\dim Z=\dim X\), and \(Y\leq Z\).