Note on weakly \(n\)-dimensional spaces (Q5938006)

Spaces considered are separable and metrizable. For a given space \(X\), \(X_{(n)}\) is the set of points in \(X\) that have arbitrarily small neighborhoods with at most \((n-1)\)-dimensional boundaries. If \(\dim X=n\), then \(\Lambda(X)=X\smallsetminus X_{(n-1)}\), and if \(n\geq 1\), one says that \(X\) is weakly \(n\)-dimensional provided that \(\dim\Lambda(X)=n-1\). The authors cite that the first example of a weakly one-dimensional space was constructed by \textit{W. Sierpiński} [Fundam. Math. 2, 81--95 (1921; JFM 48.0208.02)]. Their existence for \(n\geq 2\) was proved later [ibid. 13, 210--217 (1929; JFM 55.0975.01)] by \textit{S. Mazurkiewicz}. In this note three topics concerning weakly \(n\)-dimensional spaces are discussed: universal spaces, products, and sums. The authors prove (a) there is a universal space for the class of all weakly \(n\)-dimensional spaces, (b) (by a simpler method) Tomaszewski's result that if \(X\) is weakly \(n\)-dimensional and \(Y\) is weakly \(m\)-dimensional, then \(\dim(X\times Y)\leq n+m-1\), and (c) for each \(m\geq 1\), there exists an \(m\)-dimensional space which is not weakly \(m\)-dimensional but which admits a countable, pairwise disjoint closed cover consisting of weakly \(m\)-dimensional subspaces.