Dimensions \((m,K)\)-dim and extensions of mappings (Q388030)

The spaces in this paper are assumed to be normal and \(T_1\) and all mappings are continuous. \(\mathbb N\) denotes the set of all positive integers. In [\textit{V. V. Fedorchuk}, Mat. Vesn. 64, No. 4, 347--360 (2012; Zbl 1289.54104)] new dimensions \((m,n)\)-dim, where \(m,n \in \mathbb N\) with \(n\leq m\) were introduced such that:NEWLINENEWLINE(1) \((2,1)\)-dim=dim.NEWLINENEWLINE(2) If \((m,n)\)-dim\( X<\infty\), \(n\leq m\), then \(X\) is weakly infinite-dimensional in the sense of Smirnov.NEWLINENEWLINE(3) If \(M\) is a closed subset of X, then \((m,n)\)-dim\(M \leq (m,n)\)-dim\(X\).NEWLINENEWLINE(4) If \(X\) is hereditarily normal and \(X=X_1\cup X_2\), then \((m,n)\)-dim\( X\leq (m,n)\)-dim\( X_1+(m,n)\)-dim\(X_2+1.\)NEWLINENEWLINE(5) For every metrizable space \(X\), \((m,n)\)-dim\(X \leq\)dim\(X\).NEWLINENEWLINEIn this paper the author considers notions related to the covering dimension. He defines a more general invariant \((m,n)\)-dim\(_k\) for each triple of integers \(m, n, k\) with \(0\leq k\leq n \leq m\) and \(n\geq 1\). He proves that for any \(k_1, k_2\) and for every space \(X\) the following relation is true (Theorem 1.11): \((m,n)\)-dim\(_{k_1}X=(m,n)\)-dim\(_{k_2} X.\) Finally the late Professor Fedorchuk introduces the notion of \((m,K)\)-dim, where \(K\) is a simplicial subcomplex of the \((m-1)\)-dimensional simplex \(\Delta_{m-1}\) and proves that this dimension can be defined by means of extensions of mappings. He gives several interesting results:NEWLINENEWLINE(1) If \(K= \Delta^{n-1}_{m-1}\), then \((m,K)\)-dim\( X= (m,n)\)-dim\(_0 X.\)NEWLINENEWLINE(2) If \(K_1\subset K_2\), then \((m,K_1)\)-dim\( X\geq (m,K_2)\)-dim \(X,\) for every space \(X\).NEWLINENEWLINE(3) If \(v(\Delta_{m-1}) \subset K\subset \Delta_{m-1}\), then \((m,K)\)-dim\( X= (m,K)\)-e-dim\( X.\)NEWLINENEWLINEFor the invariant \((m,K)\)-e-dim see Definition 2.10.