Around the equality ind\(X\)=Ind\(X\) towards a unifying theorem. (English) Zbl 1030.54023
The authors solve the following problem posed by A. V. Ivanov concerning the dimension function \(\dim_0\) defined by V. V. Filippov. Problem 1. Is the sum theorem for \(\dim_0\) valid for arbitrary closed subsets?
They also prove a finite sum theorem for perfectly \(\kappa\)-normal spaces (Theorem 1) and a locally finite sum theorem for perfectly \(\kappa\)-normal spaces (Theorem 2). Using their theorems, they show the following theorem, which generalizes a result due to A. Chigogidze. Let \(X\) be a hereditarily normal perfectly \(\kappa\)-normal closure totally paracompact space. Then \(\text{ind } X = \text{Ind } X (= \text{ind}_0 X = \text{Ind}_0 X)\).
They also prove a finite sum theorem for perfectly \(\kappa\)-normal spaces (Theorem 1) and a locally finite sum theorem for perfectly \(\kappa\)-normal spaces (Theorem 2). Using their theorems, they show the following theorem, which generalizes a result due to A. Chigogidze. Let \(X\) be a hereditarily normal perfectly \(\kappa\)-normal closure totally paracompact space. Then \(\text{ind } X = \text{Ind } X (= \text{ind}_0 X = \text{Ind}_0 X)\).
Reviewer: K.Tsuda (Bunkyo-chou / Matsuyama)
MSC:
| 54F45 | Dimension theory in general topology |
Keywords:
small inductive dimension; large inductive dimension; paracompact space; prefectly \(\kappa\)-normal spaceReferences:
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