Addition and product theorems for ind. (English) Zbl 1161.54017
The authors prove versions of the Addition and Product theorems for small inductive dimension. If \(X=X_1\cup X_2\) and \(\text{ind}X_1=m\) and \(\text{ind}X_2=n\) then \(\text{ind}X\leq2(m+n+1)\); if \(X\) satisfies \(\text{ind}(A\cup B)=\max\{\text{ind}A,\text{ind}B\}\) for all pairs of closed subsets \(A\) and \(B\) then even \(\text{ind}X\leq m+n+1\).
If the formula \(\text{ind}(A\cup B)=\max\{\text{ind}A,\text{ind}B\}\) holds for all closed subsets of both \(X\) and \(Y\) that satisfy \(\text{ind}A,\text{ind}B\leq k\) then the following product theorem is obtained: if \(\text{ind}X\leq m\) and \(\text{ind}Y\leq n\) then \(\text{ind}(X\times Y)\leq m+n\) in case \(m=0\), \(n=0\) or \(m+n\leq k\); in all other cases \(\text{ind}(X\times Y)\leq2(m+n)-k-1\).
The paper also contains a finite closed sum theorem for \(\text{ind}\) as well as a number of questions pertaining to the sharpness of the estimates.
If the formula \(\text{ind}(A\cup B)=\max\{\text{ind}A,\text{ind}B\}\) holds for all closed subsets of both \(X\) and \(Y\) that satisfy \(\text{ind}A,\text{ind}B\leq k\) then the following product theorem is obtained: if \(\text{ind}X\leq m\) and \(\text{ind}Y\leq n\) then \(\text{ind}(X\times Y)\leq m+n\) in case \(m=0\), \(n=0\) or \(m+n\leq k\); in all other cases \(\text{ind}(X\times Y)\leq2(m+n)-k-1\).
The paper also contains a finite closed sum theorem for \(\text{ind}\) as well as a number of questions pertaining to the sharpness of the estimates.
Reviewer: K. P. Hart (Delft)
MSC:
| 54F45 | Dimension theory in general topology |
| 54D10 | Lower separation axioms (\(T_0\)–\(T_3\), etc.) |
| 54E35 | Metric spaces, metrizability |
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