Around the equality ind\(X\)=Ind\(X\) towards a unifying theorem (Q1398208)

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 \(\text{ind } X = \text{Ind } X (= \text{ind}_0 X = \text{Ind}_0 X)\).