Continuation of the metric dimension (Q791921)

The author introduces two ``relative'' dimension functions: given \(A\subset X\), he defines \((1)\quad X \dim(A)\leq n\) if every finite closed cover of X has a finite closed refinement \(\xi\) with the order of \(\xi\cap A\) being \(\leq n+1\); and \((2)\quad Xd(A)\leq n\) if for every system of \((n+1)\) disjoint closed sets \((B_ i,C_ i)\) in X, there is a system \(D_ i\) of closed sets, where each \(D_ i\) separates \(A_ iB_ i\) and \(A\cap \cap^{n+1}_{1}D_ i=\emptyset.\) When X is a compactum and \(A=X\), the definition (1) reduces to Alexandroff's metric dimension and (2) reduces to the metric \(d_ 2\)-dimension of Nagata-Roberts. Let dim be the usual covering dimension. The author proves (a): In hereditarily normal spaces, \[ Xd(A)\leq X \dim(A)\quad and\quad Xd(A)\geq n \] if and only if there is an essential \(f:A\to I^ n\) that extends over X. (b). In perfectly normal spaces, both \[ Xd(A)\leq \dim A\leq 2Xd(A)\quad and\quad X \dim(A)\leq \dim A\leq 2X \dim(A). \] He then constructs a certain subset A of \textit{V. V. Fedorchuk}'s space [Mat. Sb., Nov. Ser. 96(138), 41--62 (1975; Zbl 0308.54028)] and uses (b) to establish (c). There is a connected complete separable metric space A with \(1\leq \dim A\leq 2\) which is homeomorphic to its countable cartesian power \(A^{\infty}\). Finally, he also shows (d). Each n-dimensional compact Hausdorff space S has an open cover \(\xi\) with the property: if \(\{F_ i| i\in {\mathbb{Z}}\}\) is any countable family of \(\xi\)-small closed sets with \(\dim(F_ i\cap F_ j)\leq k\) for \(i\neq j\), then \[ Xd(X-\cup^{\infty}_{1}F_ i)\geq n-k-2. \]